Chapter 6

For \(x>0\), let \(h(x)=\left\\{\begin{array}{ll}\frac{1}{q}, & \text { if } x=\frac{p}{q} \\ 0, & \text { if } x \text { is irrational }\end{array}\right.\) where \(p\) and \(q>0\) are relatively prime integers, then which one of the following does not hold good? (a) \(h(x)\) is discontinuous for all \(x\) in \((0, \infty)\) (b) \(h(x)\) is continuous for each irrational in \((0, \infty)\) (c) \(h(x)\) is discontinuous for each rational in \((0, \infty)\) (d) \(h(x)\) is not derivable for all \(x\) in \((0, \infty)\)

If \(f(x)=\frac{x-e^{x}+\cos 2 x}{x^{2}}, x \neq 0\) is continuous at \(x=0\), then (a) \(f(0)=\frac{5}{2}\) (b) \([f(0)]=-2\) (c) \(\\{f(0)\\}=-0.5\) (d) \([f(0)] \cdot\\{f(0)\\}=-1.5\) (where \([x]\) and \(\\{x\\}\) denotes greatest integer and fractional part function.)

Let \([x]\) denotes the integral part of \(x \in R \cdot g(x)=x-[x]\). Let \(f(x)\) be any continuous function with \(f(0)=f(1)\), then the function \(h(x)=f(g(x))\) (a) has finitely many discontinuities (b) is discontinuous at some \(x=c\) (c) is continuous on \(R\) (d) is a constant function.

Let \(f\) be a differentiable function on the open interval \((a, b)\), Which of the following statements must be true? I. \(f\) is continuous on the closed interval \([a, b]\) II. \(f\) is bounded on the open interval \((a, b)\) III. If \(a

). Consider function \(f: R-\\{-1,1\\} \rightarrow R . f(x)=\frac{x}{1-|x|}\). Then, the incorrect statement is (a) it is continuous at the origin (b) it is not derivable at the origin (c) the range of the function is \(R\) (d) \(f\) is continuous and derivable in its dor-

The total number of points of non-differentiability of \(f(x)=\min \left[|\sin x|,|\cos x|, \frac{1}{4}\right]\) in \((0,2 \pi)\) is (a) 8 (b) 9 (c) 10 (d) 11

The function \(f(x)=[x]^{2}-\left[x^{2}\right]\) (where \([y]\) is the greatest integer less than or equal to \(y\) ) 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 \(f(x)=\left(x^{2}-1\right)\left|x^{2}-6 x+5\right|+\cos |x|\) is not differentiable at (a) \(-1\) (b) 0 (c) 1 (d) 5

Let \(f(x)= \begin{cases}\frac{1}{e^{x^{2}}}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases}\) \(f^{\prime}(0)\) is equal to (a) 0 (b) 1 (c) \(-1\) (d) Doesn't exist

Given \(f(x)=\frac{e^{x}-\cos 2 x-x}{x^{2}}\), for \(x \in R-\\{0\\}\) \(g(x)=f(\\{x\\})\), for \(n

The function \(g(x)=\left[\begin{array}{l}x+b, x<0 \\ \cos x, x \geq 0\end{array}\right.\) can be made differentiable at \(x=0\), (a) if \(b\) is equal to zero (b) if \(b\) is not equal to zero (c) if \(b\) takes any real value (d) for no value of \(b\)

The graph of function \(f\) contains the point \(P(1,2)\) and \(Q(s, r) .\) The equation of the secant line through \(P\) and \(Q\) is \(y=\left(\frac{s^{2}+2 s-3}{s-1}\right) x-1-s\). The value of \(f^{\prime}(1)\), is (a) 2 (b) 3 . (c) 4 (d) Non existent

Consider \(f(x)=\left[\frac{2\left(\sin x-\sin ^{3} x\right)+\left|\sin x-\sin ^{3} x\right|}{2\left(\sin x-\sin ^{3} x\right)-\left|\sin x-\sin ^{3} x\right|}\right] x \neq \frac{\pi}{2}\) for \(x \in(0, \pi) f(\pi / 2)=3\) where [ ] denotes the greatest integer function, then (a) \(f\) is continuous and differentiable at \(x=\pi / 2\) (b) \(f\) is continuous but not differentiable at \(x=\pi / 2\) (c) \(f\) is neither continuous nor differentiable at \(x=\pi / 2\) (d) None of the above

If \(f(x+y)=f(x)+f(y)+|x| y+x y^{2}, \forall x, y \in R\) and \(f^{\prime}(0)=0\), then (a) \(f\) need not be differentiable at every non-zero \(x\) (b) \(f\) is differentiable for all \(x \in R\) (c) \(f\) is twice differentiable at \(x=0\) (d) None of the above

Let \(g(x)=\left\\{\begin{array}{cl}3 x^{2}-4 \sqrt{x}+1, & \text { for } x<1 \\\ a x+b, & \text { for } x \geq 1\end{array}\right.\). If \(g(x)\) is the continuous and differentiable for all numbers in its domain, then (a) \(a=b=4\) (b) \(a=b=-4\) (c) \(a=4\) and \(b=-4\) (d) \(a=-4\) and \(b=4\)

Let \(f(x)\) be continuous and differentiable function for all reals. \(f(x+y)=f(x)-3 x y+f(y) .\) If \(\lim _{h \rightarrow 0} \frac{f(h)}{h}=7\), then the value of \(f^{\prime}(x)\) is (a) \(-3 x\) (b) 7 (c) \(-3 x+7\) (d) \(2 f(x)+7\)

Let \([x]\) be the greatest integer function and \(f(x)=\frac{\sin \frac{4}{4} \pi[x]}{[x]}\). Then, which one of the following does not hold good? (a) Not continuous at any point (b) Continuous at \(3 / 2\) (c) Discontinuous at 2 (d) Differentiable at \(4 / 3\)

Given, \(f(x)=\left[\begin{array}{l}b\left([x]^{2}+[x]\right)+1, \text { for } x \geq-1 \\ \sin (\pi(x+a)), \text { for } x<-1\end{array}\right.\) where \([x]\) denotes the integral part of \(x\), then for what values of \(a, b\) the function is continuous at \(x=-1\) ? (a) \(a=2 n+(3 / 2) ; b \in R ; n \in I\) (b) \(a=4 n+2 ; b \in R ; n \in I\) (c) \(a=4 n+(3 / 2) ; b \in R^{+} ; n \in I\) (d) \(a=4 n+1 ; b \in R^{+} ; n \in I\)

If both \(f(x)\) and \(g(x)\) are differentiable functions at \(x=x_{0}\), then the function defined as, \(h(x)=\) maximum \(\\{f(x), g(x)\\}\) (a) is always differentiable at \(x=x_{0}\) (b) is never differentiable at \(x=x_{0}\) (c) is differentiable at \(x=x_{0}\) when \(f\left(x_{0}\right) \neq g\left(x_{0}\right)\) (d) cannot be differentiable at \(x=x_{0}\), if \(f\left(x_{0}\right)=g\left(x_{0}\right)\)

\(\operatorname{If} f(x)=\left\\{\begin{array}{c}\frac{x \cdot \ln (\cos x)}{\ln \left(1+x^{2}\right)}, x \neq 0 \\ 0\end{array}, x=0\right.\), then (a) \(f\) is continuous at \(x=0\) (b) \(f\) is continuous at \(x=0\) but not differentiable at \(x=0\) (c) \(f\) is differentiable at \(x=0\) (d) \(f\) is not continuous at \(x=0\)

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 \(f(x)=\sqrt{1-\sqrt{1-x^{2}}}\) (a) has its domain \(-1 \leq x \leq 1\) (b) has finite one sided derivates at the point \(x=0\) (c) is continuous and differentiable at \(x=0\) (d) is continuous but not differentiable at \(x=0\)

Consider the function \(f(x)=\left|x^{3}+1\right|\). Then, (a) Domain of \(f x \in R\) (b) Range of \(f\) is \(R^{+}\) (c) \(f\) has no inverse (d) \(f\) is continuous and differentiable for every \(x \in R\)

\(f\) is a continuous function in \([a, b], g\) is a continuous function in \([b, c]\), A function \(h(x)\) is defined as \(h(x)=\left\\{\begin{array}{l}f(x) \text { for } x \in[a, b) \\ g(x) \text { for } x \in(b, c]\end{array}\right.\), If \(f(b)=g(b)\), then (a) \(h(x)\) has a removable discontinuity at \(x=b\) (b) \(h(x)\) may or may not be continuous in \([a, c]\) (c) \(h\left(b^{-}\right)=g\left(b^{+}\right)\)and \(h\left(b^{+}\right)=f\left(b^{-}\right)\) (d) \(h\left(b^{+}\right)=g\left(b^{-}\right)\)and \(h\left(b^{-}\right)=f\left(b^{+}\right)\)

Which of the following function(s) has/have the same range? (a) \(f(x)=\frac{1}{1+x}\) (b) \(f(x)=\frac{1}{1+x^{2}}\) (c) \(f(x)=\frac{1}{1+\sqrt{x}}\) (d) \(f(x)=\frac{1}{\sqrt{3-x}}\)

If \(f(x)=\sec 2 x+\operatorname{cosec} 2 x\), then \(f(x)\) is discontinuous at all points in (a) \(\\{n \pi, n \in N\\}\) (b) \(\left\\{(2 n \pm 1) \frac{\pi}{4}, n \in I\right\\}\) (c) \(\left\\{\frac{n \pi}{4}, n \in I\right\\}\) (d) \(\left\\{(2 n \pm 1) \frac{\pi}{8}, n \in I\right\\}\)

Let \(f(x)=\left\\{\begin{array}{c}x^{n} \sin \left(\frac{1}{x^{2}}\right), x \neq 0 \\ 0 \quad, x=0\end{array},(n \in I)\right.\), then (a) \(\lim _{x \rightarrow 0} f(x)\) exists for every \(n>1\) (b) \(f\) is continuous at \(x=0\) for \(n>1\) (c) \(f\) is differentiable at \(x=0\) for every \(n>1\) (d) None of the above

A function is defined as \(f(x)=\left\\{\begin{array}{c,}e^{x}, x \leq 0 \\\ |x-1|, x>0\end{array}\right.\), then \(f(x)\) is (a) continuous at \(x=0\) (b) continuous at \(x=1\) (c) differentiable at \(x=0\) (d) differentiable at \(x=1\)

A function \(f(x)\) satisfies the relation \(f(x+y)=f(x)+f(y)+x y(x+y), \forall x, y \in R\). If \(f^{\prime}(0)=-1\), then (a) \(f(x)\) is a polynomial function (b) \(f(x)\) is an exponential function (c) \(f(x)\) is twice differentiable for all \(x \in R\) (d) \(f^{\prime}(3)=8\)

If \(f(x)=\left\\{\begin{array}{cc}3 x^{2}+12 x-1, & -1 \leq x \leq 2 \\ 37-x, & 2

If \(f(x)=0\) for \(x<0\) and \(f(x)\) is differentiable at \(x=0\), then for \(x>0, f(x)\) may be (a) \(x^{2}\) (b) \(x\) (c) \(-x\) (d) \(-x^{3 / 2}\)

Statement I \(f(x)=|x| \sin x\) is differentiable at \(x=0\). Because Statement II If \(g(x)\) is not differentiable at \(x=a\) and \(h(x)\) is differentiable at \(x=a\), then \(g(x) \cdot h(x)\) cannot be differentiable at \(x=a\).

Let \(f(x)=x-x^{2}\) and \(g(x)=\\{x\\}, \forall x \in R\) where \\{\\} denotes fractional part function. Statement I \(f(g(x))\) will be continuous, \(\forall x \in R\). Because Statement II \(f(0)=f(1)\) and \(g(x)\) is periodic with period \(1 .\)

\(f(x)=\left\\{\begin{array}{lc}\sin x, & x \leq 0 \\ \tan x, 0

Number of points of discontinuity of \(\left[2 x^{3}-5\right]\) in \([1,2)\) is (where [ \(]\) ] denotes the greatest integral function.) (a) 14 (b) 13 (c) 10 (d) None of these

\(\operatorname{Max}([x],|x|)\) is discontinuous at, (a) \(x=0\) (b) \(\phi\) (c) \(x=n, n \in I\) (d) None of these

Match the entries of the following two columns.Column I \(\quad\) Column \(=\) II \hline (A) \(f(x)=\left[\begin{array}{ll}x+1, & \text { if } x<0 \\ \cos x, & \text { if } x \geq 0\end{array}\right.\) at \(x\) t \(x=0\) is (p) continuous (B) For every \(x \in R\), the function \(g(x)=\frac{\sin (\pi[x-\pi])}{1+[x]^{2}}\) (q) differentiability where \([x]\) denotes the greatest integer function is (C) \(h(x)=\sqrt{\\{x\\}^{2}}\) where \(\\{x\\}\) denotes fractional part ( \(\left.\mathrm{r}\right)\) discontinuous (r) function for all \(x \in I\), is \(k(x)=\left\\{\begin{array}{cl}x^{\frac{1}{\ln x}}, & \text { if } x \neq 1 \\\ e, & \text { if } x=1\end{array}\right.\) (D) \(k(x)=\left\\{x^{\ln x}\right.\), if \(x \neq 1\) at \(x=1\) is at \(x=1\) (s) non-derivable

\text { Match the entries of the following two columns. }\text { Column I }\text { Column II }(A) \(\lim _{x \rightarrow \infty}\left(e^{\sqrt{x^{4}+1}}-e^{\left(x^{2}+1\right)}\right)\) is (p) \(e\) (B) For \(a>0\), let \(f(x)=\left[\begin{array}{ll}\frac{a^{x}+a^{-x}-2}{x^{2}}, & \text { if } x>0 \\ 3 \ln (a-x)-2, & \text { if } x \leq 0\end{array}\right.\) (q) \(e^{2}\) If \(f\) is continuous at \(x=0\), then ' \(a\) ' equals to \(\quad\) (r) \(1 / e\) (C) Let \(L=\lim _{x \rightarrow a} \frac{x^{x}-a^{a}}{x-a}\) and \(M=\lim _{x \rightarrow a} \frac{x^{x}-a^{x}}{x-a}(a>0)\). If \(L=2 M\), then the value of ' \(a\) ' is equal to (s) non-existent

Number of points of discontinuity of \(f(x)=\tan ^{2} x-\sec ^{2} x\) in \((0,2 \pi)\) is ..........

Let \(f(x)=x+\cos x+2\) and \(g(x)\) be the inverse function of \(f(x)\), then \(g^{\prime}(3)\) equals to \(\ldots \ldots \ldots\)

Let \(f(x)=x \tan ^{-1}\left(x^{2}\right)+x^{4}\). Let \(f^{k}(x)\) denotes \(k\) th derivative of \(f(x)\) w.r.t. \(x\), \(k \in N\). If \(f^{2 m}(0) \neq 0, m \in N\), then \(m\) equals to ..........

Let \(f_{1}(x)\) and \(f_{2}(x)\) be twice differentiable functions where \(F(x)=f_{1}(x)+f_{2}(x)\) and \(G(x)=f_{1}(x)-f_{2}(x), \forall x \in R, f_{1}(0)=2\) and \(f_{2}(0)=1 .\) If \(f_{1}^{\prime}(x)=f_{2}(x)\) and \(f_{2}^{\prime}(x)=f_{1}(x), \forall x \in R\), then the number of solutions of the equation \((F(x))^{2}=\frac{9 x^{4}}{G(x)}\) is .........

Suppose the function \(f(x)-f(2 x)\) has the derivative 5 at \(x=1\) and derivative 7 at \(x=2\). The derivative of the function \(f(x)-f(4 x)-10 x\) at \(x=1\) is equal to \(\ldots \ldots \ldots .\)