Tamil Nadu Board Solutions for Chapter: Applications of Differential Calculus, Exercise 3: Exercise 3

Using the Lagrange’s mean value theorem determine the values of $x$ at which the tangent is parallel to the secant line at the end points of the given interval:

$f\left(x\right)=\left(x-2\right)\left(x-7\right), x\in \left[3,11\right]$

Show that the value in the conclusion of the mean value theorem for $f\left(x\right)=\frac{1}{x}$ on a closed interval of positive numbers $\left[a,b\right]$ is $\sqrt{ab}$.

Show that the value in the conclusion of the mean value theorem for $f\left(x\right)=A{x}^2+Bx+C$ on any interval $\left[a,b\right]$ is $\frac{a+b}{2}$.

A race car driver is racing at ${20}^{th}$ km. If his speed never exceeds $150$ km/hr, what is the maximum distance he can cover in the next two hours.

Suppose that for a function $f\left(x\right), f^{\text{'}}\left(x\right)\le 1$ for all $1\le x\le 4$. Show that$f\left(4\right)-f\left(1\right)\le 3$.

Does there exist a differentiable function $f\left(x\right)$ such that $f\left(0\right)=-1, f\left(2\right)=4$ and $f^{\text{'}}\left(x\right)\le 2$ for all $x$. Justify your answer.

Show that there lies a point on the curve $f\left(x\right)=x\left(x+3\right)e^{-\frac{\mathrm{\pi }}{2}}, -3\le x\le 0$ where tangent drawn is parallel to the $x$-axis.

Using mean value theorem prove that for, $a>0, b>0, \left|e^{-a}-e^{-b}\right|<\left|a-b\right|$.