Applications of Derivatives Questions (514)

52. A sector is to be cut from a circular piece of wire of perimeter 20 metres (i.e., \(r + r + r\theta = 20\) metres). The maximum area (in m²) of the sector is:

For the function $f(x)=\sin x+3x-\dfrac{2}{\pi}(x^2+x)$, where $x\in\left[0,\dfrac{\pi}{2}\right]$, consider the following two statements: (I) $f$ is increasing in $\left(0,\dfrac{\pi}{2}\right)$. (II) $f'$ is decreasing in $\left(0,\dfrac{\pi}{2}\right)$. Between the above two statements:

If the function $f(x)=\left(\dfrac{1}{x}\right)^{2x}$, $x>0$, attains the maximum value at $x=\dfrac{1}{e}$, then:

Let $f(x) = \begin{cases} -x^3 + a, & 0 \leq x < 1 \\ x, & 1 \leq x \leq 3 \end{cases}$:

The curve $y=f(x)$ satisfies $\dfrac{d^2y}{dx^2}=6x-4$ and has a local minimum value $5$ at $x=1$. $f(0)$ is

Let the function $f(x)=2x^3+(2p-7)x^2+3(2p-9)x-6$ have a maxima for some value of $x<0$ and a minima for some value of $x>0$. Then the set of all values of p is:

If the equation of the normal to the curve $y=\dfrac{x-a}{(x+b)(x-2)}$ at the point $(1,-3)$ is $x-4y=13$, then the value of $a+b$ is equal to ___.

If the functions $f(x)=\dfrac{x^3}{3}+\dfrac{ax^2}{2}+2bx$ and $g(x)=\dfrac{x^3}{3}+bx^2+ax$, $a\neq2b$, have a common extreme point, then $a+2b+7$ is equal to:

If $f(x)=x^2+g'(1)x+g''(2)$ and $g(x)=f(1)x^2+xf'(x)+f''(x)$, then the value of $f(4)-g(4)$ is equal to ___.

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that $f'(x)+f(x)=\displaystyle\int_0^2 f(t)\,dt$. If $f(0)=e^{-2}$, then $2f(0)-f(2)$ is equal to ___.

The sum of the absolute maximum and minimum values of the function $f(x)=|x^2-5x+6|-3x+2$ in the interval $[-1,3]$ is equal to:

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length \(x\). The maximum area enclosed by the park is

Let $f(x)$ be a cubic polynomial such that its local maximum is at $(0,1)$ and local minimum at $(1,-2)$, and $f(-1)<0$, $f(2)>0$. The value of $\sin^{-1}(\cos[\alpha])$ may be equal to (where $\alpha$ is a root of $f(x)=0$, and $[\cdot]$ is the greatest integer function)

Given function is \(f(x) = \cot^{-1}x + x\); \(D_f = \mathbb{R}\). Which of the following is true?