Embibe Experts Solutions for Exercise 4: EXERCISE 5.5

 Verify Rolle's theorem for the function $f\left(x\right)=x(x-1{)}^2$ in the interval $[0,1]$.

 Verify Rolle's theorem of the function $f\left(x\right)=e^x\cos x$ in the interval $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$.

Verify Lagrange's mean value theorem for the function $f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(x-3\right)$ in $\left[0,4\right]$.

Verify the hypothesis and conclusion of Lagrange's mean value Theorem for the function :

$f\left(x\right)=\frac{1}{4x-1}; 1 \le x \le 4.$

Find $c$ of the mean value theorem for the function $f\left(x\right)=\frac{1}{4x-1},1\le x\le 4$.

Find a point on the curve $y=x^3$, where the tangent to the curve is parallel to the chord joining the points $\left(1, 1\right)$ and $\left(3,27\right)$.

For a function $f\left(x\right), f\left(0\right)=0$ and $f^{\text{'}}\left(x\right)=\frac{1}{1+x^2}$, prove that $f\left(x\right)+f\left(y\right)=f\left(\frac{x+y}{1-xy}\right)$.

Let $f\left(x\right)=x^n,n$ being a positive integer. The value of $\text{'}n\text{'}$ for which the equality $f^{\text{'}}\left(a+b\right)=f^{\text{'}}\left(a\right)+f^{\text{'}}\left(b\right)$ is valid for all $a,b>0$ is