Chapter 14

Let \(f\) be \(a\) continuous, diferentiable and bijective function. If the tangent to \(y=f(x)\) at \(x=a\) is also the normal to \(y=f(x)\) at \(x=b\), then there exists at least one \(c \in(a, b)\) such that (A) \(f^{\prime}(c)=0\) (B) \(f^{\prime}(c)>0\) (C) \(f^{\prime}(c)<0\) (D) None of these

The values of \(k\) for which the function \(f(x)=k x^{3}-9 x^{2}+9 x+3\) may be increasing on \(R\) are (A) \(k>3\) (B) \(k<3\) (C) \(k \leq 3\) (D) None of these

The least possible value of \(k\) for which the function \(f(x)=x^{2}+k x+1\) may be increasing on \([1,2]\) is (A) 2 (B) \(-2\) (C) 0 (D) None of these

Let \(a+b=4, a<2\) and \(g(x)\) be a monotonically increasing function of \(x\). Then, \(f(a)=\int_{0}^{a} g(x) d x+\int_{0}^{b} g(x) d x\) (A) increases with increase in \((b-a)\) (B) decreases with increase in \((b-a)\)

The equation \(x+e^{x}=0\) has (A) only one real root (B) only two real roots (C) no real root (D) None of these

If \(f^{\prime \prime}(x)<0 \forall x \in(a, b)\), then \(f^{\prime}(x)=0\) (A) exactly once in \((a, b)\) (B) at most once in \((a, b)\) (C) at least once in \((a, b)\) (D) None of these

The minimum value of \(a \tan ^{2} x+b \cot ^{2} x\) equals the maximum value of \(a \sin ^{2} \theta+b \cos ^{2} \theta\) where \(a>b>0\), when (A) \(a=b\) (B) \(a=2 b\) (C) \(a=3 b\) (D) \(a=4 b\)

If \(f(x)=\frac{x^{2}-1}{x^{2}+1}\), for every real number, then minimum value of \(f\) (A) Does not exist (B) Is note attained even through \(f\) is bounded (C) Is equal to 1 (D) Is equal to \(-1\)

If \(y=a \log |x|+b x^{2}+x\) has its extremum values at \(x=-1\) and \(x=2\), then (A) \(a=2, b=-1\) (B) \(a=2, b=-1 / 2\) (C) \(a=-2, b=1 / 2\) (D) None of these

If \(f(x)\) and \(g(x)\) are differentiable functions for \(0 \leq x \leq 1\) such that \(f(0)=2, g(0)=0, f(1)=6, g(1)=2\), then in the interval \((0,1)\), (A) \(f^{\prime}(x)=0\) for all \(x\) (B) \(f^{\prime}(x)=2 g^{\prime}(x)\) for at least one \(x\) (C) \(f^{\prime}(x)=2 g^{\prime}(x)\) for at most one \(x\) (D) None of these

The difference between the greatest and least values of the function \(f(x)=\cos x+\frac{1}{2} \cos 2 x-\frac{1}{3} \cos 3 x\) is (A) \(2 / 3\) (B) \(8 / 7\) (C) \(9 / 4\) (D) \(3 / 8\)

The range of values of \(k\) for which the function \(f(x)=\left(k^{2}-7 k+12\right) \cos x+2(k-4) x+\log 2\) does not possess critical points, is (A) \((1,5)\) (B) \((1,5)-\\{4\\}\) (C) \((1,4)\) (D) None of these

If a differentiable function \(f(x)\) has a relative minimum at \(x=0\), then the function \(y=f(x)+a x+b\) has a relative minimum at \(x=0\) for (A) all \(a>0\) (B) all \(b>0\) (C) all \(a\) and \(b\) (D) \(\mathrm{all} b\) if \(a=0\)

Let \(f(x)\) and \(g(x)\) be defined and differentiable for \(x \geq x_{0}\) and \(f\left(x_{0}\right)=g\left(x_{0}\right), f^{\prime}(x)>g^{\prime}(x)\) for \(x>x_{0}\), then (A) \(f(x)x_{0}\) (b) \(f(x)=g(x), x>x_{0}\) (C) \(f(x)>g(x), x>x_{0}\) (d) None of these

If \(\alpha\) and \(\beta(\alpha<\beta)\) be two different real roots of the equation \(a x^{2}+b x+c=0\), then (A) \(\alpha>-\frac{b}{2 a}\) (B) \(\beta<-\frac{b}{2 a}\) (C) \(\alpha<-\frac{b}{2 a}<\beta\) (D) \(\beta<-\frac{b}{2 a}<\alpha\)

If \(p(x)=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}\) and \(|p(x)| \leq\left|e^{x-1}-1\right|\) for all \(x \geq 0\), then \(\left|a_{1}+2 a_{2}+3 a_{3}+\ldots+n a_{n}\right|\) (A) \(\leq 1\) (B) \(\geq 1\) (C) \(\geq 0\) (D) \(\leq 0\)

The maximum value of radius vector where \(\frac{c^{4}}{r^{2}}=\frac{a^{2}}{\sin ^{2} t}+\frac{b^{2}}{\cos ^{2} t} ;(a, b>0)\) is (A) \((a+b)^{2}\) (B) \(\frac{c^{4}}{(a+b)^{2}}\) (C) \(\frac{c^{2}}{a+b}\) (D) \(c^{2}(a+b)\)

Let \(f(x)=\left\\{\begin{array}{cc}-x^{3}+\log _{2} b & 0

The second drivative \(f^{\prime \prime}(x)\) of the function \(f(x)\) exists for all \(x\) in \([0,1]\) and satisfies \(\left|f^{\prime \prime}(x)\right| \leq 1 .\) If \(f(0)=f(1)\), then for all \(x\) in \([0,1]\) (A) \(\left|f^{\prime}(x)\right|<1\) (B) \(\left|f^{\prime}(x)\right|>1\) (C) \(\left|f^{\prime}(x)\right|=1\) (D) \(f(x)\) is constant

Let the function \(f\) be defined as \(f(x)=\left\\{\begin{aligned} \frac{P(x)}{x-2}, & x \neq 2 \\ 7, & x=2 \end{aligned}\right.\) where \(P(x)\) is a polynomial such that \(P^{\prime \prime \prime}(x)\) is identically equal to 0 and \(P(3)=9 .\) If \(f(x)\) is continuous at \(x=2\), then (A) \(P(x)=2 x^{2}-x-6\) (B) \(P(x)=2 x^{2}+x-6\) (C) \(P(x)=2 x^{2}-x+6\) (D) None of these

The equation \(x^{5}-3 x-1=0\) has, in the interval \([1,2]\) (A) at least one root (B) at most one root (C) no root (D) a unique root

If the equation \(x-\sin x=k\) has a unique root in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), then the range of values of \(k\) are (A) \(\left(1-\frac{\pi}{2}, \frac{\pi}{2}-1\right)\) (B) \(\left[1-\frac{\pi}{2}, \frac{\pi}{2}-1\right]\) (C) \(\left[0, \frac{\pi}{2}+1\right]\) (D) None of these

The largest term in the sequence \(a_{n}=\frac{n}{n^{2}+10}, n \in N\) is (A) \(\frac{4}{26}\) (B) \(\frac{3}{19}\) (C) \(\frac{7}{18}\) (D) None of these

The range of values of \(a\) for which all roots of the equation \(3 x^{4}+4 x^{3}-12 x^{2}+a=0\) are real and distinct is (A) \((0,5)\) (B) \((1,4)\) (C) \((-1,5)\) (D) None of these

If \(\phi(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x)<0\) in \((-1,1)\), then \(\phi(x)\) strictly increases in the interval (A) \(\left(0, \frac{1}{2}\right)\) (B) \(\left(\frac{1}{2}, 1\right)\) (C) \((-1,0)\) (D) \((0,1)\)

\(f(x)\) is a cubic function with \(f(1)=-6, f(-1)=10\) and has maxima at \(x=-1\). If \(f^{\prime}(x)\) has minima at \(x=1\), then (A) \(f(x)=x^{3}+3 x^{2}-9 x+5\) (B) \(f(x)=x^{3}-3 x^{2}-9 x+5\) (C) \(f(x)=x^{3}-3 x^{2}+9 x+5\) (D) \(f(x)=x^{3}-3 x^{2}-9 x+5\)

If the function \(f(x)= \begin{cases}-x^{3}+\frac{b^{3}-b^{2}+b-1}{b^{2}+3 b+2}, & 0 \leq x<1 \\ 2 x-3, & 1 \leq x \leq 3\end{cases}\) has the least value at \(x=1\), then all possible real values of \(b\) are (A) \((-1,1)\) (B) \((-2,-1) \cup[1, \infty)\) (C) \((-2,1)\) (D) None of these

The function \(f(x)=\frac{|x+1|}{x^{2}}\) is strictly decreasing in the interval (A) \((-\infty,-2) \cup(0,1)\) (B) \((-2,0) \cup(1, \infty)\) (C) \((-2,-1) \cup(0, \infty)\) (D) None of these

The range of values of \(a\) so that the equation \(x^{3}-3 x+\) \(a=0\) has three real and distinct roots is (A) \((-\infty,-2) \cup(2, \infty)\) (B) \((-2,0)\) (C) \((-2,0)\) (D) \((-2,2)\)

The curves \(\frac{x^{2}}{a}+\frac{y^{2}}{b}=1\) and \(\frac{x^{2}}{a_{1}}+\frac{y^{2}}{b_{1}}=1\) will cut orthogonally if (A) \(a+b=a_{1}+b_{1}\) (B) \(a-b=a_{1}-b_{1}\) (C) \(\frac{1}{a}-\frac{1}{b}=\frac{1}{a_{1}}-\frac{1}{b_{1}}\) (D) None of these

The point on the curve \(3 x^{2}-4 y^{2}=72\) which is nearest to the line \(3 x+2 y+1=0\) is (A) \((6,-3)\) (B) \((6,3)\) (C) \((-6,3)\) (D) \((-6,-3)\)

If the function \(f(x)=\left(a^{2}-3 a+2\right) \cos \frac{x}{2}+(a-1) x\) possesses critical points, then \(a\) belongs to the interval (A) \((-\infty, 0) \cup(4, \infty)\) (B) \((-\infty, 0] \cup[4, \infty)\) (C) \((-\infty, 0] \cup\\{1\\} \cup[4, \infty)\) (D) None of these

If the function \(f(x)=\int_{0}^{x} \mid \log _{2}\left(\log _{3}\left(\log _{4}(\cos t\right.\right.\) \(+a\) ) ) \(\mid d t\), be increasing for all real values of \(x\), then (A) \(a \geq 2\) (B) \(a \geq 5\) (C) \(a<5\) (D) \(a<2\)

The value of \(n\), for which the function \(f(x)=\left(x^{2}-4\right)^{n}\) \(\left(x^{2}-x+1\right), n \in N\) assumes a local minima at \(x=2\), is (A) an even number (B) an odd number (C) an irrational number (D) cannot be determined

If the function \(f(x)=\left(1-\frac{\sqrt{21-4 b-b^{2}}}{b+1}\right) x^{3}\) \(+5 x+\sqrt{16}\) increases for all \(x\), then (A) \(b \in(-1,2)\) (B) \(b \in(-7,3)-\\{-1\\}\) (C) \(b \in(-7,-1) \cup(2,3)\) (D) None of these

Let \(f(x)=(x-3)(x-4)(x-4)(x-5)(x-6)\), then (A) \(f^{\prime}(x)\) has four roots (B) three roots of \(f^{\prime}(x)=0\) lie in \((3,4) \cup(4,5) \cup(5,6)\) (C) the equation \(f^{\prime}(x)=0\) has only one root (D) three roots of \(f^{\prime}(x)=0\) lie in \((2,3) \cup(3,4) \cup(4,5)\)

Let \(f^{\prime \prime}(x)>0 \forall x \in R\) and \(g(x)=f(2-x)+f(4+x)\). Then, \(g(x)\) is increasing in (A) \((-\infty,-1)\) (B) \((-\infty, 0)\) (C) \((-1, \infty)\) (D) None of these

The curves \(x^{2}-4 y^{2}+c=0\) and \(y^{2}=4 x\) will cut orthogonally for (A) \(c \in(0,16)\) (B) \(c \in(-3,4)\) (C) \(c \in(3,4)\) (D) None of these

Which of the following is not true? The function \(f(x)=x^{2}+\frac{\lambda}{x}\) has a (A) minimum at \(x=2\) if \(\lambda=16\) (B) maximum at \(x=2\) if \(\lambda=16\) (C) maximum for no real value of \(\lambda\) (D) point of inflexion at \(x=1\) if \(\lambda=-1\)

If the parabola \(y=f(x)\), having axis parallel to the \(y\)-axis, touches the line \(y=x\) at \((1,1)\), then (A) \(2 f^{\prime}(0)+f(0)=1\) (B) \(2 f(0)+f^{\prime}(0)=1\) (C) \(2 f(0)-f^{\prime}(0)=1\) (D) \(2 f^{\prime}(0)-f(0)=1\)

The angle between the tangents at any point \(P\) and the line joining \(P\) to the origin \(O\), where \(P\) is a point on the curve \(\ln \left(x^{2}+y^{2}\right)=c \tan ^{-1} \frac{y}{x}, c\) is a constant (A) varies as \(\tan ^{-1} x\) (B) varies as \(\tan ^{-1} y\) (C) is a constant (D) None of these

If the equation \(a x^{2}+b x+c=0\) has two distinct positive roots, then the equation \(a x^{2}+(b+6 a) x+\) \((c+3 b)=0\) has (A) two positive roots (B) exactly one positive root (C) at least one positive root (D) no positive root

If \(f(x)\) is continuous in \([a, b]\) and differentiable in \((a, b)\) then there exists at least one \(c \in(a, b)\) such that \(\frac{f(b)-f(a)}{b^{3}-a^{3}}\) equals (A) \(3 c^{2} f^{\prime}(\mathrm{C})\) (B) \(\frac{f^{\prime}(c)}{3 c^{2}}\) (C) \(f(c) f^{\prime}(C)\) (D) None of these

Let \(f(x)=\ln x\) and \(g(x)=x^{2} .\) If \(c \in(4,5)\), then \(c \ln \left(\frac{4^{25}}{5^{16}}\right)\) equals (A) \(c \ln 5-8\) (B) \(2\left(c^{2} \ln 4-8\right)\) (C) \(2\left(c^{2} \ln 5-8\right)\) (D) \(c \ln 4-8\)

\(\mathrm{f} 0\frac{\sin x}{x}\) (B) \(\frac{2}{\pi}<\frac{\sin x}{x}\) (C) \(\frac{\sin x}{x}<1\) (D) \(\frac{\sin x}{x}>1\)

If \(0\cos x\) (B) \(\cos (\sin x)<\cos x\) (C) \(\cos (\sin x)>\sin (\cos x)\) (D) \(\cos (\sin x)<\sin (\cos x)\)

The function \(f(x)=|x+2|+|x-1|\) is (A) increasing in \((1, \infty)\) (B) increasing in \([1, \infty)\) (C) decreasing in (-\infty, - 2] (D) decreasing in \((-\infty,-2)\)

If \(g(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x)<0\) for \(0 \leq x \leq 1\), then (A) \(g(x)\) increases in \(\left(-\infty, \frac{1}{2}\right)\) (B) \(g(x)\) increases in \(\left(0, \frac{1}{2}\right)\) (C) \(g(x)\) decreases in \(\left(\frac{1}{2}, 1\right)\) (D) \(g(x)\) decreases in \(\left(\frac{1}{2}, \infty\right)\)

The function \(f(x)=\frac{|x-1|}{x^{2}}\) (A) increases in \((-\infty, 0) \cup(1,2)\) (B) increases in \((0,1) \cup(2, \infty)\) (C) decreases in \((0,1) \cup(2, \infty)\) (D) decreases in \((-\infty, \infty) \cup(1,2)\)

Let \(h(x)=f(x)-[f(x)]^{2}+[f(x)]^{3}\) for every real number \(x\). Then (A) \(h\) is increasing whenever \(f\) is increasing (B) \(h\) is increasing whenever \(f\) is decreasing (C) \(h\) is decreasing whenever \(f\) is decreasing (D) nothing can be said in general