Chapter 12

If \(f(x)=\left\\{\begin{array}{l}3, x<0 \\ 2 x+1, x \geq 0\end{array}\right.\), then (A) both \(f(x)\) and \(f(|x|)\) are differentiable at \(x=0\) (B) \(f(x)\) is differentiable but \(f(|x|)\) is not differentiable at \(x=0\) (C) \(f(|x|)\) is differentiable but \(f(x)\) is not differentiable at \(x=0\) (D) both \(f(x)\) and \(f(|x|)\) are not differentiable at \(x=0\)

Let \(f(x)=\cos x\) and \(g(x)=[x+2]\), where \([.]\) denotes the greatest integer function. Then, \((\text { gof })^{\prime}\left(\frac{\pi}{2}\right)\) is (A) 1 (B) 0 (C) \(-1\) (D) does not exist

Let \(f(x)=\left\\{\begin{array}{cl}\frac{1}{|x|} & |x| \geq 1 \\ a x^{2}+b & |x|<1\end{array}\right.\). If \(f(x)\) is continuous and differentiable at any point, then (A) \(a=\frac{1}{2}, b=-\frac{3}{2}\) (B) \(a=-\frac{1}{2}, b=\frac{3}{2}\) (C) \(a=1, b=-1\) (D) None of these

The function \(f(x)=[x] \cos \left(\frac{2 x-1}{2}\right) \pi\), where \([.]\) denotes the greatest integer function, is discontinuous at (A) all \(x\) (B) all integer points (C) no \(x\) (D) \(x\) which is not an integer

The left-hand derivative of \(f(x)=[x] \sin (\pi x)\) at \(x=k\), \(k\) an integer and \([x]=\) greatest integer \(\leq x\), is (A) \((-1)^{k}(k-1) \pi\) (B) \((-1)^{k-1} \cdot(k-1) \pi\) (C) \((-1)^{k} \cdot k \pi\) (D) \((-1)^{k-1} \cdot k \pi\).

If \(\lim _{x \rightarrow a+} f(x)=l=\lim _{x \rightarrow a-} g(x)\) and \(\lim _{x \rightarrow a-} f(x)=m=\) \(g(x)\), then the function \(f(x) \cdot g(x)\) (A) is not continuous at \(x=a\) (B) has a limit when \(x \rightarrow a\) and it is equal to \(l \mathrm{~m}\) (C) is continuous at \(x=a\) (D) has a limit when \(x \rightarrow a\) but it is not equal to \(l \mathrm{~m}\)

Let \([x]\) denotes the greatest integer less than or equal to \(x\). If \(f(x)=[x \sin \pi x]\), then \(f(x)\) is (A) continuous at \(x=0\) (B) continuous in \((-1,0)\) (C) differentiable at \(x=1\) (D) differentiable in \((-1,1)\)

The function \(f(x)=[x]^{2}-\left[x^{2}\right]\) (where \([x]\) is the greatest integer less than or equal to \(x\) ), is discontinuous at (A) all integers (B) all integers except 0 and 1 (C) all integers except 0 (D) all integers except 1

The function \(f(x)=[x]^{2}-\left[x^{2}\right]\) (where \([x]\) is the greatest integer less than or equal to \(x\) ), is discontinuous at (A) all integers (B) all integers except 0 and 1 (C) all integers except 0 (D) all integers except 1

Let \(f(x)\) be a function satisfying the condition \(f(-x)=f(x)\), for all real \(x\). If \(f^{\prime}(0)\) exists, then its value is (A) 0 (B) 1 (C) \(-1\) (D) None of these

If \(f(x)=\left\\{\begin{array}{cc}\frac{x\left(3 e^{1 / x}+4\right)}{2-e^{1 / x}}, x \neq 0 \\ 0, x=0\end{array}\right.\), then \(f(x)\) is (A) continuous as well as differentiable at \(x=0\) (B) continuous but not differentiable at \(x=0\) (C) differentiable but not continuous at \(x=0\) (D) None of these

The function \(f(x)=\frac{1}{u^{2}+u-2}\), where \(u=\frac{1}{x-1}\), is discontinuous at the points (A) \(x=-2,1, \frac{1}{2}\) (B) \(x=\frac{1}{2}, 1,2\) (C) \(x=1,0\) (D) None of these

Let \(f(x)=[3+2 \cos x], x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)\), where \([.]\) denotes the greatest integer function. The number of points of discontinuity of \(f(x)\) is (A) 3 (B) 2 (C) 5 (D) None of these

The set of points of continuity of the function \(f(x)=\sqrt{\frac{1}{2}-\cos ^{2} x}\) is (A) \(\left\\{x: \frac{\pi}{4}+2 n \pi \leq x \leq \frac{3 \pi}{4}+2 n \pi, n \in I\right\\}\) (B) \(\left\\{x: \frac{5 \pi}{4}+2 n \pi \leq x \leq \frac{7 \pi}{4}+2 n \pi, n \in I\right\\}\) (C) \(\begin{aligned}\left\\{x: \frac{\pi}{4}+2 n \pi \leq\right.& \left.x \leq \frac{3 \pi}{4}+2 n \pi\right\\} \\ & \cup\left\\{x: \frac{5 \pi}{4}+2 n \pi \leq x \leq \frac{7 \pi}{4}+2 n \pi\right\\} \end{aligned}\) (D) None of these

The function \(f(x)=\operatorname{are} \tan \frac{1}{x-5}\) has (A) discontinuity of the first kind at \(x=5\) (B) discontinuity of the second kind at \(x=5\) (C) removable discontinuity at \(x=5\) (D) continuous at \(x=5\)

If \(f(x)=\sum_{k=0}^{n} a_{k}|x-1|^{k}\), where \(a_{i} \in R\) then (A) \(f(x)\) is continuous at \(x=1\) for all \(a_{k} \in R\) (B) \(f(x)\) is differentiable at \(x=1\) for all \(a_{k} \in R\) (C) \(f(x)\) is differentiable at \(x=1\), provided \(a_{2 k+1}=0\) (D) \(f(x)\) is continuous at \(x=1\), provided \(a_{2 k}=0\)

If \(f(x)=\cos ^{-1}\left(\frac{2 x}{1+x^{2}}\right)\), then \(f(x)\) is differentiable on (A) \((-\infty, \infty)\) (B) \((-\infty, \infty) \backslash\\{0\\}\) (C) \((-\infty, \infty) \backslash\\{-1,1\\}\) (D) None of these

The set of points of discontinuities of the function \(f(x)=\sqrt{x}-[\sqrt{x}]\), where \([x]\) denotes the greatest inte- ger less than or equal to \(x\), contains the set (A) \((-\infty, 0)\) (B) \(\left\\{n^{2}: n \in N\right\\}\) (C) \(N\) (D) \(\\{2 n: n \in N\\}\)

If \(f(x)=|3-x|+(3+x)\), where \((x)\) denotes the least integer greater than or equal to \(x\), then (A) \(f(x)\) is continuous as well as differentiable at \(x=3\) (B) \(f(x)\) is continuous but not differentiable at \(x=3\) (C) \(f(x)\) is differentiable but not continuous at \(x=3\) (D) \(f(x)\) is neither differentiable nor continuous at \(x=3\)

Let \(f(x)=\left\\{\begin{array}{cl}\frac{1+\cos x}{(\pi-x)^{2}} \cdot \frac{\sin ^{2} x}{\log \left(1+\pi^{2}-2 \pi x+x^{2}\right)} & , x \neq \pi \\\ k & , x=\pi\end{array}\right.\) If \(f(x)\) is continuous at \(x=\pi\), then \(k\) is equal to (A) \(\frac{1}{4}\) (B) \(\frac{1}{2}\) (C) \(\frac{-1}{2}\) (D) \(\frac{-1}{4}\)

Let \(f(x)\) be a continuous function defined for \(1 \leq x \leq 3\). If \(f(x)\) takes rational values for all \(x\) and \(f(2)=10\), then \(f(1.5)\) is equal to (A) 0 (B) 10 (C) not defined (D) any constant

If \(f(x)=\int_{0}^{x} t \cos \frac{1}{t} d t\), then the number of points of discontinuity of \(f(x)\) in the interval \((0, \pi)\) is (A) 1 (B) 2 (C) 0 (D) None of these

If \(f(x)=(-1)^{\left[x^{2}\right]}\), where \([.]\) denotes the greatest integer function, then (A) \(f(x)\) is discontinuous for \(x=n^{1 / 3}\), where \(n \in I\) (B) \(f(3 / 2)=1\) (C) \(f^{\prime}(x)=0\) for \(-1

If \(f(x)=\left[\frac{1}{\sqrt{2}}(\cos x+\sin x)\right], 0

Let \(f(x)=a[x]+b e^{|x|}+c|x|^{2}\), where \(a, b\) and \(c\) are real constants. If \(f(x)\) is differentiable at \(x=0\), then (A) \(b=0, c=0, a \in R\) (B) \(a=0, c=0, b \in R\) (C) \(a=0, b=0, c \in R\) (D) None of these

If \(f(x)=[x] \sin \left(\frac{\pi}{[x+1]}\right)\), where \([.]\) denotes the greatest integer function, then the points of discontinuity of \(f\) in the domain are (A) \(Z\) (B) \(Z \backslash\\{0\\}\) (C) \(R \backslash[-1,0)\) (D) None of these

The value of \(f(0)\) so that the function \(f(x)=\frac{\sqrt[3]{1+x}-\sqrt[4]{1+x}}{x}\) becomes continuous at \(x=0\), is (A) \(\frac{1}{12}\) (B) \(\frac{7}{12}\) (C) 0 (D) None of these

If \(f\) is an even function such that \(\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}\) has some finite non-zero value, then (A) \(f\) is continuous and derivable at \(x=0\) (B) \(f\) is continuous but not derivable at \(x=0\) (C) \(f\) may be discontinuous at \(x=0\) (D) None of these

If \(f\) is differentiable function satisfying \(f(0)=0\), and if \(g(x)=\frac{f(x)}{x}\), then the value, that should be assigned to \(g(0)\), so that \(g\) is continuous at ' 0 ' is (A) \(]\) (B) 0 (C) \(f(0)\) (D) \(f^{\prime}(0)\)

Let \(f(x)=\frac{1}{[\sin x]},[.]\) being the greatest integer function, then (A) \(f(x)\) is not continuous, where \(x \in(2 n \pi, 2 n \pi+\pi), n \in I\) (B) \(f(x)\) is differentiable at \(x=\frac{\pi}{4}\) (C) \(f(x)\) is differentiable at \(x=\frac{\pi}{2}\) (D) None of these

The function \(f(x)=\left\\{\begin{array}{cc}1-2 x+3 x^{2}-4 x^{3}+\ldots \text { to } & \infty, x \neq-1 \\ 1 & , x=-1\end{array}\right.\) is (A) continuous and derivable at \(x=-1\) (B) neither continuous nor derivable at \(x=-1\) (C) continuous but not derivable at \(x=-1\) (D) None of these

If \(f(x)=x+\frac{x}{1+x}+\frac{x}{(1+x)^{2}}+\ldots\) to \(\infty\), then at \(x=0\), \(f(x)\) (A) has no limit (B) is discontinuous (C) is continuous but not differentiable (D) is differentiable

The function \(f(x)=\left[x^{2}\right]+[-x]^{2}\), where [.] denotes the greatest integer function, is (A) continuous and derivable at \(x=2\) (B) neither continuous nor derivable at \(x=2\) (C) continuous but not derivable at \(x=2\) (D) None of these

If the function \(f(x)=\left\\{\begin{array}{cl}(1-|\tan x|)^{\frac{a}{\left.\right|^{\tan x \mid}}} & , \frac{-\pi}{4}

If \(f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}(\log a)^{n}\), then at \(x=0, f(x)\) (A) has no limit (B) is discontinuous (C) is continuous but not differentiable (D) is differentiable

The values of constants \(a\) and \(b\) so as to make the function \(f(x)=\left\\{\begin{array}{ll}\frac{1}{|x|},|x| \geq 1 \\ a x^{2}+b,|x|<1\end{array}\right.\) continuous as well as differentiable for all \(x\), are (A) \(a=\frac{-1}{2}, b=\frac{3}{2}\) (B) \(a=\frac{1}{2}, b=\frac{3}{2}\) (C) \(a=\frac{-1}{2}, b=\frac{-3}{2}\) (D) None of these

If \(f(x)=\left[\tan ^{2} x\right]\) (where \([.]\) denotes the greatest integer function), then (A) \(\lim _{x \rightarrow 0} f(x)\) does not exist (B) \(f(x)\) is continuous at \(x=0\) (C) \(f(x)\) is non-differentiable at \(x=0\) (D) \(f(0)=1\).

The values of \(p\) and \(q\) for which the function \(f(x)= \begin{cases}\frac{\sin (p+1) x+\sin x}{x}, & x<0 \\ q, & x=0 \\\ \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^{3 / 2}}, & x=0\end{cases}\) is continuous for all \(x\) in \(R\), are (A) \(p=\frac{1}{2}, q=\frac{3}{2}\) (B) \(p=\frac{1}{2}, q=-\frac{3}{2}\) (C) \(p=\frac{5}{2}, q=\frac{1}{2}\) (D) \(p=-\frac{3}{2}, q=\frac{1}{2}\)

If \(f: R \rightarrow R\) is a fucntion defined by \(f(x)=[x] \cos\) \(\left(\frac{2 x-1}{2}\right) \pi\), where \([x]\) denotes the greatest integer fucntion, then \(f\) is (A) continuous for every real \(x\) (B) discontinuous only at \(x=0\) (C) discontinuous only at non-zero integral values of \(x\) (D) continuous only at \(x=0\)

Consider the fucntion, \(f(x)=|x-2|+|x-5|, x \in R\) Statement-1: \(f^{\prime}(4)=0\) Statement-2: \(f\) is continuous in \([2,5]\) differentiable in \((2,5)\) and \(f(2)=f(5)\) (A) Statement- 1 is false, Statement- 2 is true (B) Statement-1 is true, statement- 2 is true; statement- 2 is a correct explanation for Statement-1 (C) Statement- 1 is true, statement- 2 is true; statement- 2 is not a correct explanation for statement- 1 (D) Statement- 1 is true, statement- 2 is false

If \(f(x)=|x|+[x-1]\), where \([.]\) is greatest integer function, then \(f(x)\) is: (A) continuous at \(x=0\) as well as at \(x=1\) (B) continous at \(x=0\) but not at \(x=1\) (C) continuous at \(x=1\) but not at \(x=0\) (D) neither continuous at \(x=0\) nor at \(x=1\)

Amongst the following functions, a function that is differentiable at \(x=0\) is (A) \(\cos (|x|)-|x|\) (B) \(\cos (|x|)+|x|\) (C) \(\sin (|x|)+|x|\) (D) \(\sin (|x|)-|x|\)

Let \(f(x)=x^{2}-8 x+12, x \in[2,6]\). Statement-1: \(f^{\prime}(c)=0\) for some \(c \in(2,6)\) Statement- \(\mathbf{2}: f\) is continuous on \([2,6]\) and differentiable on \((2,6)\) with \(f(2)=f(6)\) (A) Statement-1 is true, Statement-2 is true, Statement- 2 is a correct explanantion for Statement- 1 (B) Statement-1 is true, Statement-2 is true, Statement- 2 is not a correct expalantion for Statement-1 (C) Statement- 1 is true, Statement- 2 is false (D) Statement- 1 is false, Statement- 2 is true

Let \(f(x)=x|x|\) and \(g(x)=\sin x\). Statement-1: gof is differentiable at \(x=0\) and its derivative is continuous at that point. Statement- \(\mathbf{2}\) : gof is twice differentiable at \(x=0\). (A) Statement-1 is true, Statement- 2 is true; Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is true, Statement-2 is true; Statement- 2 is not a correct explanation for Statement- 1 (C) Statement- 1 is true, Statement- 2 is false (D) Statement- 1 is false, Statement- 2 is true

The function \(f(x)=\left\\{\begin{array}{l}(x+1)^{2-\left(\frac{1}{|x|}+\frac{1}{x}\right)}, x \neq 0 \text { is } \\ 0 & , x=0\end{array}\right.\) (A) discontinuous at only one point (B) discontinuous exactly at two points (C) continuous everywhere (D) None of these

\(f(x)=\left\\{\begin{array}{cc}\frac{e^{[x]+|x|}-2}{[x]+|x|}, & x \neq 0 \\\ -1, & x=0\end{array},([.]\right.\) denotes the greatest integer function), then (A) \(f(x)\) is continuous at \(x=0\) (B) \(\lim _{x \rightarrow 0^{-}} f(x)=-1\) (C) \(\lim _{x \rightarrow 0^{-}} f(x)=1\) (D) None of these

The Dirichlet function, defined as \(f(x)=\left\\{\begin{array}{l}1 \text { if } x \text { is rational } \\ 0 \text { if } x \text { is irrational }\end{array}\right.\), is (A) continuous for all real \(x\) (B) continuous only at some values of \(x\) (C) discontinuous for all real \(x\) (D) discontinuous only at some values of \(x\)

Let \(f: R \rightarrow R\) be a function such that \(f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3}, f(0)=0\) and \(f^{\prime}(0)=3\) Then, (A) \(f(x)\) is a quadratic function (B) \(f(x)\) is continuous but not differentiable (C) \(f(x)\) is differentiable in \(R\) (D) \(f(x)\) is bounded in \(R\)

If \(f(x)=\left\\{\begin{array}{l}x, \text { when } x \text { is rational } \\\ 1-x, \text { when } x \text { is irrational }\end{array}\right.\), then (A) \(f(x)\) is continuous for all real \(x\) (B) \(f(x)\) is discontinuous for all real \(x\) (C) \(f(x)\) is continuous only at \(x=1 / 2\) (D) \(f(x)\) is discontinuous only at \(x=1 / 2\).

The points where the function \(f(x)=[x]+|1-x|,-1\) \(\leq x \leq 3\), where \([.]\) denotes the greatest integer function, is not differentiable, are (A) \(x=-1,0,1,2,3\) (B) \(x=-1,0,2\) (C) \(x=0,1,2,3\) (D) \(x=-1,0,1,2\)