Applications of Derivatives

If the maximum value of $a$, for which the function $f_a(x)=\tan^{-1}2x-3ax+7$ is non-decreasing in $\left(-\dfrac\pi6,\dfrac\pi6\right)$, is $\bar a$, then $f_{\bar a}\left(\dfrac\pi8\right)$ is equal to

Suppose $y=y(x)$ be the solution curve to the differential equation $\dfrac{dy}{dx}-y=2-e^{-x}$ such that $\displaystyle\lim_{x\to\infty}y(x)$ is finite. If $a$ and $b$ are respectively the $x$- and $y$-intercepts of the tangent to the curve at $x=0$, then the value of $a-4b$ is equal to ______.

Let a function $f:\mathbf R\to\mathbf R$ be defined as :

$$f(x)=\begin{cases}\displaystyle\int_0^x(5-|t-3|)\,dt, & x>4\\ x^2+bx, & x\le4\end{cases}$$

where $b\in\mathbf R$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

Let M and N be the number of points on the curve $y^5-9xy+2x=0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value of $M+N$ equals ______.

Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area $A_1$ is twice the area $A_2$. Then the normal to the curve perpendicular to the line $2x-12y=15$ does NOT pass through the point.

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is $\tan^{-1}\dfrac34$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is ______.

If the minimum value of $f(x)=\dfrac{5x^2}2+\dfrac\alpha{x^5}$, $x>0$, is 14, then the value of $\alpha$ is equal to :

The minimum value of the twice differentiable function $f(x)=\displaystyle\int_0^xe^{x-t}f'(t)\,dt-\left(x^2-x+1\right)e^x$, $x\in\mathbf R$, is :

Let $f:[0,1]\to\mathbf R$ be a twice differentiable function in (0, 1) such that $f(0)=3$ and $f(1)=5$. If the line $y=2x+3$ intersects the graph of $f$ at only two distinct points in (0, 1), then the least number of points $x\in(0,1)$, at which $f''(x)=0$, is ______.

The function $f(x)=xe^{x(1-x)}$, $x\in\mathbf R$, is :

The sum of the absolute maximum and absolute minimum values of the function $f(x)=\tan^{-1}(\sin x-\cos x)$ in the interval $[0,\pi]$ is :

Let $f(x)=3^{(x^2-2)^3}+4$, $x\in\mathbf R$. Then which of the following statements are true ?

P : $x=0$ is a point of local minima of $f$

Q : $x=\sqrt2$ is a point of inflection of $f$

R : $f'$ is increasing for $x>\sqrt2$

If the tangent to the curve $y=x^3-x^2+x$ at the point (a, b) is also tangent to the curve $y=5x^2+2x-25$ at the point (2, −1), then |2a + 9b| is equal to _______.

The number of integral values of b for which $2x^3-3x^2-12x+b=0$ has three distinct real roots is _______.

An air duct is made of a thick metallic sheet. The duct is open from front and back. Its cross-section is a trapezium of base angles 60° and area 1.5 m$^2$. Its length is 1 m. (see the figure). If its height $x=x_0$ (in metres) so that the area of the metallic sheet used for its construction is minimum, then $16(x_0)^4$ is equal to _______.

Let A(0, a + 2), B(0, a), C(−2, 0) and D(2, 0) be four points and the lines AD and BC intersect at P(x, y). If the locus of P is the curve f(x, y) = 0 and the tangent at the point (4, γ) on this curve is $\dfrac x\alpha+\dfrac y\beta=1$, then α (β − γ) is equal to _______.

Let $f: \mathbb R \to \mathbb R$ be defined by
$$f(x) = \frac{x^2 - 3x - 6}{x^2 + 2x + 4}$$
Then which of the following statements is (are) TRUE?

Let $f: \left[-\frac\pi2, \frac\pi2\right] \to \mathbb R$ be a continuous function such that
$$f(0) = 1 \quad\text{and}\quad \int_0^{\pi/3} f(t)\,dt = 0$$
Then which of the following statements is (are) TRUE?

The radius of the circle $C$ is ____.

The value of $2m_1 + 3n_1 + m_1n_1$ is ____.