Applications of Derivatives

If $f$ and $g$ are differentiable functions in $[0,1]$ satisfying $f(0)=2=g(1)$, $g(0)=0$ and $f(1)=6$, then for some $c\in\,]0,1[$ :

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha\log|x|+\beta x^2+x$ then :

If the Rolle's theorem holds for the function $f(x)=2x^3+ax^2+bx$ in the interval $[-1,1]$ for the point $c=\dfrac12$, then the value of $2a+b$ is :

If $f(x)=\left(\dfrac35\right)^x+\left(\dfrac45\right)^x-1$, $x\in\mathbf R$, then the equation $f(x)=0$ has :

For the curve $y=3\sin\theta\cos\theta$, $x=e^\theta\sin\theta$, $0\le\theta\le\pi$, the tangent is parallel to $x$-axis when $\theta$ is :

Two ships A and B are sailing straight away from a fixed point O along routes such that $\angle AOB$ is always 120°. At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/hr) :

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $=\sqrt3$ is :

Let $f$ and $g$ be two differentiable functions on $\mathbb R$ such that $f'(x)>0$ and $g'(x)<0$, for all $x\in\mathbb R$. Then for all $x$ :

If the volume of a spherical ball is increasing at the rate of 4π cc/sec, then the rate of increase of its radius (in cm/sec), when the volume is 288π cc, is :

The area of the region above the $x$-axis bounded by the curve $y=\tan x$, $0\le x\le\dfrac\pi2$ and the tangent to the curve at $x=\dfrac\pi4$ is :

Let $f:\left[\frac12,1\right]\to\mathbb R$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f'(x)<2f(x)$ and $f\left(\frac12\right)=1$. Then the value of $\displaystyle\int_{1/2}^1f(x)\,dx$ lies in the interval

The number of points in $(-\infty,\infty)$, for which $x^2-x\sin x-\cos x=0$, is

Let $f(x)=x\sin\pi x$, $x>0$. Then for all natural numbers $n$, $f'(x)$ vanishes at

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are

A vertical line passing through the point $(h,0)$ intersects the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$ at the points $P$ and $Q$. Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$. If $\Delta(h)=$ area of the triangle $PQR$, $\Delta_1=\displaystyle\max_{1/2\le h\le1}\Delta(h)$ and $\Delta_2=\displaystyle\min_{1/2\le h\le1}\Delta(h)$, then $\dfrac8{\sqrt5}\Delta_1-8\Delta_2=$

The function $f(x)=2|x|+|x+2|-\big||x+2|-2|x|\big|$ has a local minimum or a local maximum at $x=$

Which of the following is true for $0<x<1$?

If the function $e^{-x}f(x)$ assumes its minimum in the interval $[0,1]$ at $x=\frac14$, which of the following is true?

A line $L:y=mx+3$ meets $y$-axis at $E(0,3)$ and the arc of the parabola $y^2=16x$, $0\le y\le6$ at the point $F(x_0,y_0)$. The tangent to the parabola at $F(x_0,y_0)$ intersects the $y$-axis at $G(0,y_1)$. The slope $m$ of the line $L$ is chosen such that the area of the triangle $EFG$ has a local maximum.

Match List I with List II and select the correct answer using the code given below the lists:

The function $f:[0,3]\to[1,29]$, defined by $f(x)=2x^3-15x^2+36x+1$, is