Applications of Derivatives Questions (514)

In the figure, $\theta_1+\theta_2=\dfrac{\pi}{2}$ and $\sqrt{3}(BE)=4(AB)$. If the area of $\triangle CAB$ is $2\sqrt{3}-3$ unit$^2$, when $\dfrac{\theta_2}{\theta_1}$ is the largest, then the perimeter (in unit) of $\triangle CED$ is equal to _______.

Let $f(x)=x^3+x^2f'(1)+2xf''(2)+f'''(3)$, $x\in\mathbb{R}$. Then the value of $f'(5)$ is:

Let $A$ be the region enclosed by the parabola $y^2=2x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $A$ is ________.

Let $g(x)=f(x)+f(1-x)$ and $f''(x)>0,\ x\in(0,1)$. If $g$ is decreasing in the interval $(0,\alpha)$ and increasing in the interval $(\alpha,1)$, then $\tan^{-1}(2\alpha)+\tan^{-1}\!\left(\dfrac{1}{\alpha}\right)+\tan^{-1}\!\left(\dfrac{\alpha+1}{\alpha}\right)$ is equal to

A wire of length 20 m is to be cut into two pieces. A piece of length $\ell_1$ is bent to make a square of area $A_1$ and the other piece of length $\ell_2$ is made into a circle of area $A_2$. If $2A_1+3A_2$ is minimum, then $(\pi\ell_1):\ell_2$ is equal to:

Let $x=2$ be a local minima of the function $f(x)=2x^4-18x^2+8x+12$, $x\in(-4,4)$. If M is the local maximum value of the function f in $(-4,4)$, then M =

Let a function $f$ be defined as $f(x) = \begin{cases} \frac{|x-1|}{x^2+1} & \text{if } x > -1 \\ x^2 & \text{if } x \leq -1 \end{cases}$. Then the number of critical point(s) on the graph of this function is/are:

Let $f(x)=2x+\tan^{-1}x$ and $g(x)=\log_e\!\left(\sqrt{1+x^2}+x\right)$, $x\in[0,3]$. Then:

Let $f:(0,1)\to\mathbb{R}$ be a function defined by $f(x)=\dfrac{1}{1-e^{-x}}$, and $g(x)=(f(-x)-f(x))$. Consider two statements: (I) g is an increasing function in $(0,1)$; (II) g is one-one in $(0,1)$. Then,

We have \(f(x) = \alpha\log(x) + \beta x^2 + x\). If \(f'(-1) = 0\) and \(f'(2) = 0\), find \(\alpha\) and \(\beta\).The equations are \(\alpha + 2\beta - 1 = 0\) and \(\alpha + 8\beta + 2 = 0\). Which values satisfy these?

A wire of length 36cm is cut into two pieces. One piece is bent into a square and the other into an equilateral triangle. The minimum value of the total area is

Angle between the tangents to the curve \(y = x^2 - 5x + 6\) at the points \((2, 0)\) and \((3, 0)\) is

The equation of the tangent to the curve \(y = x + \dfrac{4}{x^2}\), that is parallel to the x-axis, is

If $x_1$ and $x_2$ are the abscissae of the two points on $y=x(1-x^2)$ at which tangent lines are parallel to the chord joining $(1,0)$ and $(-1,0)$, then $x_1^2+x_2^2=$