G Tewani Solutions for Chapter: Applications of Derivatives, Exercise 4: DPP

If $f\left(x\right)$ and $g\left(x\right)$ are continuous and differentiable functions, then prove that there exists $c\in [a, b]$ such that

$\frac{f^{\text{'}}\left(c\right)}{f\left(a\right)-f\left(c\right)}+\frac{g^{\text{'}}\left(c\right)}{g\left(b\right)-g\left(c\right)}=1$.

Consider $f\left(x\right)=|1-x|, 1\le x\le 2$ and $g\left(x\right)=f\left(x\right)+b\sin \left(\frac{\pi }{2}x\right), 1\le x\le 2,$ then which of the following is correct?

If $c=\frac{1}{2}$ and $f\left(x\right)=2x-x^2,$ then interval of $x$ in which $\text{LMVT}$ is applicable, is

If a twice differentiable function $f\left(x\right)$ on $\left(a, b\right)$ and continuous on $[a, b]$ is such that $f^{\text{'}\text{'}}\left(x\right)<0$ for all $x\in (a, b)$

then for any $c\in \left(a, b\right), \frac{f\left(c\right)-f\left(a\right)}{f\left(b\right)-f\left(c\right)}>$

Let $a,n\in N$ such that $a\ge n^3.$ Then $\sqrt[3]{a+1}-\sqrt[3]{a}$ is always:

Given $f\text{'}\left(1\right)=1$ and $f\left(2x\right)=f\left(x\right),\forall x>0$. If $f\text{'}\left(x\right)$ is differentiable then there exists a number $c\in \left(2, 4\right)$ such that $f\text{'}\text{'}\left(c\right)$ equals:

Let $f\left(x\right)=\left|\begin{array}{ccc}1& 1& 1\\ 3-x& 5-3x^2& 3x^3-1\\ 2x^2-1& 3x^5-1& 7x^8-1\end{array}\right|.$ Then which of the following is/are correct?

Which of the following is correct?