Chapter 5

Find the mean rate of change of the following functions in the interval \([1,2]\) :- i. \(\quad f(x)=x^{2}\). \\{Ans. 3\\} ii. \(\quad f(x)=x^{3}\). \\{Ans. 7\(\\}\) iii. \(f(x)=\sqrt{x}\). \\{Ans. 0.414\\} iv. \(\quad f(x)=\frac{1}{x} .\) \\{ns. \(\left.-0.5\right\\}\) v. \(\quad f(x)=e^{x}\). \\{Ans. 4.67\\} vi. \(f(x)=\ln x .\\{\) Ans. \(0.693\\}\) vii. \(f(x)=\sin x .\\{\) Ans. \(0.068\\}\) viii. \(f(x)=\cos x\). \\{ns. \(-0.956\\}\)

For what value of \(a\), the mean rate of change of the function \(f(x)=x^{2}\) in the interval \([a, a+1]\) is 2 ?

Find the instantaneous rate of change of the following functions at \(x=1\) and also find Stationary points:- i. \(\quad f(x)=x^{2}\) ii. \(\quad f(x)=x^{3}\) iii. \(f(x)=\sqrt{x}\) iv. \(f(x)=\frac{1}{x}\). v. \(f(x)=e^{x}\). vi. \(f(x)=\ln x\). vii. \(f(x)=\sin x\) viii. \(f(x)=\cos x\)

For what value of \(a\), the mean rate of change of the function \(f(x)=x^{3}\) in the interval \([-1, a]\) is equal to the instantaneous rate of change at \(a ?\)

Find the approximate value of the following:- i. \(\cos 31^{\circ}\). ii. \(\log 10.21\) iii. \(\sqrt[5]{33}\). iv. \(\cot 45^{\circ} 10^{\prime}\)

Test the following functions for monotonicity and find stationary points:- i. \(f(x)=2 x^{3}-9 x^{2}+12 x+29\). ii. \(\quad y=x^{3}-3 x^{2}+6 x-17\). iii. \(f(x)=x^{3}-3 x\). iv. \(\quad y=x^{2}(x-3)^{2}\). v. \(\quad f(x)=x^{9}+3 x^{7}+64\) v. \(\quad f(x)=x^{9}+3 x^{7}+64\) viii. \(f(x)=\frac{x}{\ln x}\) ix. \(\quad f(x)=x^{x}\). x. \(\quad f(x)=x^{\frac{1}{x}}\). xi. \(\quad f(x)=x e^{x}\). xii. \(f(x)=x e^{-x}\) xiii. \(f(x)=e^{\frac{1}{x}}\). xiv. \(f(x)=x e^{\frac{1}{x}}\) xv. \(\quad f(x)=\log _{x}(\ln x)\). xvi. \(\quad f(x)=\ln (1+x)-\frac{2 x}{2+x}\). xvii. \(f(x)=x+\sin x\). xviii. \(f(x)=\frac{4 \sin x-2 x-x \cos x}{2+\cos x}, \quad 0

Show that the function \(f(x)=\sin x, \quad x\) is a rationalno. \(=x, \quad x\) is an irrationalno. has positive derivative at \(x=0\) but \(f(x)\) is not increasing at \(x=0\).

Show that the function \(\begin{aligned} f(x) &=\frac{x}{2}+x^{2} \sin \frac{1}{x}, \quad x \neq 0 \\ &=0, \quad x=0 \end{aligned}\) is continuous and differentiable in any neighbourhood of \(x=0\) and \(f^{\prime}(0)\) is positive but \(f(x)\) is not increasing at \(x=0\).

If \(g(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x)<0 ; 0 \leq x \leq 1\), show that \(g(x)\) increases in \(0

Given \(g(x)=f\left(x^{2}-x-10\right)+f\left(14+x-x^{2}\right)\) and \(f^{\prime \prime}(x)>0\) for all real \(x\), except at finite no. of real values of \(x\) for which \(f^{\prime \prime}(x)=0\). Discuss the monotonicity of the function \(g(x)\). \\{ns. \((\infty,-3)\) decreasing, \(\left(-3, \frac{1}{2}\right)\) increasing, \(\left(\frac{1}{2}, 4\right)\) decreasing, \((4, \infty)\) increasing \(\\}\)

Investigate the following functions for extremum at \(x=0:-\) i. \(\quad f(x)=\sin x-x\). ii. \(\quad f(x)=\sin x-x+\frac{x^{3}}{3 !}\). iii. \(f(x)=\sin x-x+\frac{x^{3}}{3 !}-\frac{x^{4}}{4 !}\). iv. \(\quad f(x)=e^{\frac{1}{x}}, \quad x \neq 0\) \(=0, \quad x=0 .\) v. \(f(x)=\cosh x+\cos x\). vi. \(f(x)=\cos x-1+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}\). vii. \(f(x)=\cos x-1+\frac{x^{2}}{2}\). viii. \(f(x)=x+x^{\frac{2}{3}}\). ix. \(f(x)=x^{2}+x^{\frac{1}{3}}\). x. \(f(x)=x^{\frac{4}{3}}+2 .\) xi. \(f(x)=x^{\frac{5}{3}}-3 .\)

Test the following functions for extremum:- i. \(f(x)=2 x^{3}-15 x^{2}-84 x+8\). ii. \(\quad f(x)=x^{3}-6 x^{2}+9 x-8\). iii. \(\quad f(x)=-\frac{3}{4} x^{4}-8 x^{3}-\frac{45}{2} x^{2}+105\). iv. \(\quad f(x)=\frac{3}{4} x^{4}-x^{3}-9 x^{2}+7\) v. \(\quad f(x)=x^{4}-8 x^{3}+22 x^{2}-24 x+12\). vi. \(\quad f(x)=x(x+1)^{3}(x-3)^{2}\) vii. \(\quad f(x)=\frac{x^{2}-3 x+2}{x^{2}+2 x+1}\). viii. \(f(x)=3 \sqrt[3]{x^{2}}-x^{2}\). ix. \(f(x)=\sqrt[3]{(x-1)^{2}}+\sqrt[3]{(x+1)^{2}}\). x. \(f(x)=-2 x, \quad x<0\) \(=3 x+5, \quad x \geq 0 .\\{\) xi. \(\begin{aligned} f(x) &=2 x^{2}+3, \quad x \neq 0 \\ &=4, \quad x=0 .\\{\text { Ans. maxima at } 0\\} \end{aligned}\) \(f(x)=\frac{50}{3 x^{4}+8 x^{3}-18 x^{2}+60}\) \(f(x)=\sqrt{e^{x^{2}}-1}\) \(f(x)=x e^{x}\) \(f(x)=x^{4} e^{-x^{2}}\) \(f(x)=x^{2} e^{-x}\) xvii. \(\quad f(x)=\frac{4 x}{x^{2}+4}\) xviii. \(\quad f(x)=-x^{2} \sqrt[5]{(x-2)^{2}}\) xix. \(\quad f(x)=\frac{14}{x^{4}-8 x^{2}+2}\) \(f(x)=\sqrt[3]{2 x^{3}+3 x^{2}-36 x}\) xx. \(\quad f(x)=\sqrt[3]{2 x^{3}+3 x^{2}-36 x}\). xxi. \(\quad f(x)=x^{2} \ln x\) xxii. \(\quad f(x)=x \ln ^{2} x\) xxiv. \(\quad f(x)=|x|+|x-1|+|x-2|\). xxv. \(\quad f(x)=\sin ^{4} x+\cos ^{4} x, 0

The function \(f(x)=a \sin x+\frac{1}{3} \sin 3 x\) has maximum value at \(x=\frac{\pi}{3}\). Find the value of \(a\).

Given \(f(x)=|x-2|+\ln \left(a^{2}-1\right), \quad x<2\) \(=3 x+5, \quad x \geq 2 .\) Find values of \(a\) for which \(f(x)\) has local minima at \(x=2\).

Find the polynomial of degree 6 which satisfies \(\lim _{x \rightarrow 0}\left(1+\frac{J(x)}{x^{3}}\right)^{x}=e^{2}\) and has local maxima at \(x=1\) and local minima at \(x=0 \& x=2\).

Find greatest \& least value of the following functions in the indicated intervals:- i. \(\quad f(x)=x^{3}-3 x\) in \([0,2]\). ii. \(\quad f(x)=2 x^{3}-3 x^{2}-12 x+1\) in \(\left[-2, \frac{5}{2}\right]\). iii. \(\quad f(x)=2 x^{3}-24 x+107\) in \([1,3]\). iv. \(\quad f(x)=x^{2} \ln x\) in \([1, e] .\) v. \(\quad f(x)=\sqrt{\left(1-x^{2}\right)\left(1+2 x^{2}\right)}\) in \([-1,1]\). vi. \(\quad f(x)=\cos ^{-1}\left(x^{2}\right)\) in \(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\). vii. \(\quad f(x)=x+\sqrt{x}\) in \([0,4]\).

Prove that:- i. \(\quad e^{x}>1+x, \quad x \neq 0\). ii. \(\quad x-\frac{x^{3}}{6}<\sin x0\). iii. \(\frac{x}{1+x} \leq \ln (1+x) \leq x, \quad x>-1\). iv. \(\frac{x}{1+x^{2}}<\tan ^{-1} x0\). v. \(\quad \ln x>\frac{2(x-1)}{x+1}, \quad x>1\). vi. \(\quad 2 x \tan ^{-1} x \geq \ln \left(1+x^{2}\right)\). vii. \(\ln (1+x)>\frac{\tan ^{-1} x}{1+x}, \quad x>0\). viii. \(\sin x0\). ix. \(\quad \sin x+\tan x>2 x, \quad 01+\frac{x^{2}}{2}, \quad x \neq 0\). xi. \(1+x \ln \left(x+\sqrt{x^{2}+1}\right) \geq \sqrt{1+x^{2}}\). xii. \(\quad 2 \sin x+\tan x \geq 3 x\) for \(0 \leq x<\frac{\pi}{2}\).

The function \(f(x)=x^{4}-62 x^{2}+a x+9\) attains its maximum value on the interval \([0,2]\) at \(x=1\). Find the value of \(a\). =

Let \(f(x)=-x^{3}+\frac{b^{3}-b^{2}+b-1}{b^{2}+3 b+2}, \quad 0 \leq x<1\) \(=2 x-3, \quad 1 \leq x \leq 3 .\) Find all possible real values of \(b\) such that \(f(x)\) has the smallest value at \(x=1\).

Use the function \(f(x)=x^{\frac{1}{x}}, x>0\) to determine the bigger of the two numbers \(e^{\pi} \& \pi^{e}\).

Find the intervals of concavity of the following functions:- i. \(f(x)=x^{4}+x^{3}-18 x^{2}+24 x-12\). ii. \(f(x)=3 x^{5}-5 x^{4}+3 x-2\). iii. \(f(x)=x^{6}-10 x^{4}\). iv. \(f(x)=\ln \left(x^{2}-1\right)\). v. \(f(x)=(x+1)^{4}+e^{x}\). vi. \(f(x)=x^{2} \ln x .\) vii. \(f(x)=x+x^{\frac{4}{3}}\). viii. \(f(x)=x+x^{\frac{5}{3}}+1\). ix. \(f(x)=x+x^{\frac{2}{3}}\). x. \(\quad f(x)=x^{2}, \quad x \leq 0\) \(=x^{3}, \quad x>0\). \(=x^{2}, \quad x>0\). \(=x^{2}, \quad x>1\).

Test the indicated points for point of inflection:- i. \(f(x)=x^{3}-5 x^{2}+3 x-5\) at \(x=1, \frac{5}{3}, 2\). ii. \(f(x)=x^{4}-12 x^{3}+48 x^{2}\) at \(x=1,2,3,4\). iii. \(f(x)=x+x^{\frac{5}{3}}-2\) at \(x=0\). iv. \(f(x)=x^{2}+x^{\frac{4}{3}}+1\) at \(x=0\). v. \(f(x)=x+x^{\frac{2}{3}}+4\) at \(x=0\). vi. \(f(x)=x+x^{\frac{3}{5}}-3\) at \(x=0\). vii. \(f(x)=\sin x+\frac{x^{3}}{3 !}-\frac{x^{5}}{5 !}\) at \(x=0\). viii. \(f(x)=e^{x}-\frac{x^{2}}{2}-\frac{x^{3}}{6}\) at \(x=0\). ix. \(\quad f(x)=\sin x, \quad x \geq 0\) \(=x-\frac{x^{3}}{6}, \quad x<0\) at \(x=0\). \\{Ans. 0 is point of inflection\\}

Test the following functions for concavity \(\&\) find points of inflection:- i. \(\quad f(x)=x+36 x^{2}-2 x^{3}-x^{4}\). ii. \(\quad f(x)=3 x^{4}-8 x^{3}+6 x^{2}+12\). iii. \(f(x)=(x+2)^{6}+2 x+2\). iv. \(f(x)=\frac{x}{1+x^{2}}\). v. \(f(x)=\ln \left(1+x^{2}\right)\). vi. \(\quad f(x)=x^{4}(12 \ln x-7)\). vii. \(\quad f(x)=x \ln x\). viii. \(f(x)=\frac{\ln x}{x}\). ix. \(f(x)=x^{x}\). x. \(f(x)=x e^{x}\).

Verify Rolle's theorem for the following functions:- i. \(\quad f(x)=2 x^{3}+x^{2}-4 x-2\) in \([-\sqrt{2}, \sqrt{2}]\). ii. \(f(x)=\sin x\) in \([0, \pi]\). iii. \(f(x)=\tan x\) in \([0, \pi]\). iv. \(f(x)=\cos \frac{1}{x}\) in \([-1,1]\). v. \(\quad f(x)=x(x+3) e^{-\frac{x}{2}}\) in \([-3,0]\). vi. \(\quad f(x)=e^{x}(\sin x-\cos x)\) in \(\left[\frac{\pi}{4}, \frac{5 \pi}{4}\right]\). vii. \(f(x)=|x|\) in \([-1,1]\). viii. \(f(x)=3+(x-2)^{\frac{2}{3}}\) in \([1,3]\). ix. \(\quad f(x)=\ln \left(\frac{x^{2}+a b}{(a+b) x}\right)\) in \([a, b], a>0 .\) x. \(f(x)=(x-a)^{m}(x-b)^{n}\) in \([a, b]\), where \(m\) and \(n\) are positive integers.

Show that the equation \(x^{3}-3 x+c=0\) cannot have two different roots in the interval \((0,1)\).

Prove that the equation \(3 x^{5}+15 x-8=0\) has only one real solution.

Prove that the polynomial \(x^{4}-4 x-1\) has exactly two different real roots.

Show that the equation \(x e^{x}=2\) has only one solution which lies in the interval \((0,1)\).

Show that the equation \(x^{4}+2 x-2=0\) has exactly one real solution in the interval \((0,1)\).

If the equation \(a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots \ldots+a_{1} x=0\) has a positive solution \(a\), then prove that the equation \(n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\ldots \ldots+a_{1}=0\) also has a positive solution which is smaller than \(a\).

If \(2 a+3 b+6 c=0\), then show that the equation \(a x^{2}+b x+c=0\) has at least one real root between 0 and \(1 .\)

Let \(\frac{a_{0}}{n+1}+\frac{a_{1}}{n}+\frac{a_{2}}{n-1}+\ldots \ldots \ldots+\frac{a_{n-1}}{2}+\frac{a_{n}}{1}=0 .\) Show that there exists at least one real \(x\) between 0 and 1 such that \(a_{0} x^{n}+a_{1} x^{n-1}+\ldots \ldots+a_{n-1} x+a_{n}=0\)

Show that \(f(x)=x^{2}\) satisfies Lagrange's Mean value theorem in the interval \([0,1]\) and find the value of \(c\).

Prove the validity of Lagrange's theorem for the function \(y=\ln x\) in the interval \([1, e]\) and find the value of \(c\).

With the aid of Lagrange's theorem prove the inequalities \(\frac{a-b}{a} \leq \ln \frac{a}{b} \leq \frac{a-b}{b}\), for the condition \(0