Vinod Singh and Shweta Pawar Solutions for Chapter: Applications of Derivatives, Exercise 3: Competitive Thinking

The point on the curve $6y=x^3+2$ at which $y$-co-ordinate is changing $8$ times as fast as $x$-co-ordinate is ____________

If $f\left(x\right)$ satisfies the conditions of Rolle's theorem in $\left[1,2\right]$  and $f\left(x\right)$ is continuous in $\left[1,2\right]$, then${\int }_1^2{f}^{\text{'}}\left(x\right)dx$  is equal to

Let $f\left(x\right)=x^3+b{x}^2+cx+d, 0<b^2<c$. Then, $f$

Let $p\left(x\right)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p\left(1\right)=6$ and $p\left(3\right)=2$, then $p^{\text{'}}\left(0\right)$ is

Let $f\left(x\right)$ be a polynomial of degree four having extreme values at $x=1$ and $x=2$. If $\underset{x\to 0}{\lim }\left[1+\frac{f\left(x\right)}{x^2}\right]=3$, then $f\left(2\right)$ is equal to

If $A>0, B>0$ and $A+B=\frac{\mathrm{\pi }}{3}$, then the maximum value of $\tan A\tan B$ is

A wire of length $2$ units is cut into two parts which are bent respectively to form a square of side$=x$ units and a circle of radius$=r$ units. If the sum of the areas of the square and the circle so formed is minimum, then

Consider $f\left(x\right)={\tan }^{-1} \left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), x\in \left(0,\frac{\pi }{2}\right)$. A normal to $y=f\left(x\right)$ at $x=\frac{\pi }{6}$ also passes through the point