Embibe Experts Solutions for Chapter: Continuity and Differentiability, Exercise 4: EXERCISE-4

Let $f:\left[a,b\right]\to R$ be continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$. If $f\left(a\right)<f\left(b\right)$, then show that $f^{\text{'}}\left(c\right)>0$ for some $c\in \left(a,b\right)$.

Show that the function $f\left(x\right)=x^n+px+q$ cannot have more than two real roots if $n$ is even and more than three if $n$ is odd.

Prove that if $f$  is differentiable on $[a, b]$ and if $f\left(a\right)=f\left(b\right)=0$ then for any real $\alpha$ there is an $x\in (a,b)$ such that $\alpha$ $f\left(x\right)+f^{\text{'}}\left(x\right)=0$

Let $f$ be continuous on $\left[a,b\right]$ and differentiable on $(a,b)$. If $f\left(a\right)=a$ and $f\left(b\right)=b$, show that there exist distinct $c_1,c_2$, in $(a, b)$ such that $f^{\text{'}}\left(c_1\right)+f^{\text{'}}\left(c_2\right)=2$

Prove the inequality ${\mathrm{e}}^x>(1+x)$ using LMVT for all $x\in R_o$ and use it to determine which of the two numbers ${\mathrm{e}}^{\pi }$ and ${\pi }^e$ is greater.

Suppose that on the interval $[-2,4]$ the function $f$ is differentiable, $f\left(-2\right)=1$ and $\left|f^{\text{'}}\right(x\left)\right|\le 5.$ Find the bounding function of f on $[-2,4]$, using LMVT.

Using LMVT prove that $\tan x>x$ in $\left(0,\frac{\pi }{2}\right)$.

Using LMVT prove that $\sin x<x$ for $x>0$.