Applications of Derivatives

Consider the function $f:(0,\infty)\to(-\infty,\infty)$ given by
$$f(x)=\sqrt x\,\log_e(x)-x+1.$$
Then which one of the following statements is TRUE ?

Let $y=f(x)$ be the real valued function defined on the interval $(0,\infty)$, satisfying $y(1)=0$ and the differential equation
$$x\frac{dy}{dx}=y-x^3.$$
Then which of the following statements is (are) TRUE ?

The number of critical points of the function $f(x)=\begin{cases}\left|\dfrac{\sin x}x\right|, & x\ne0\\[4pt]1, & x=0\end{cases}$ in the interval $(-2\pi,2\pi)$ is equal to :

Let $f(x)$ be a polynomial of degree 5, and have extrema at $x=1$ and $x=-1$. If $\displaystyle\lim_{x\to0}\left(\frac{f(x)}{x^3}\right)=-5$, then $f(2)-f(-2)$ is equal to :

$\displaystyle\max_{0\le x\le\pi}\left(16\sin\left(\frac x2\right)\cos^3\left(\frac x2\right)\right)$ is equal to:

Consider the system of linear equations in $x,y,z$:
$$x+2y+tz=0,\qquad 6x+y+5tz=0,\qquad 3x+t^2y+f(t)z=0,$$
where $f:\mathbb R\to\mathbb R$ is a differentiable function. If this system has infinitely many solutions for all $t\in\mathbb R$, then $f$

Let $f:\mathbb R\to\mathbb R$ be a differentiable function such that $f\left(\dfrac{x+y}3\right)=\dfrac{f(x)+f(y)}3$ for all $x,y\in\mathbb R$, and $f'(0)=3$. Then the minimum value of the function $g(x)=3+e^xf(x)$, is:

Let $A=\begin{bmatrix}1&3&-1\\2&1&\alpha\\0&1&-1\end{bmatrix}$ be a singular matrix. Let $f(x)=\displaystyle\int_0^x\left(t^2+2t+3\right)dt$, $x\in[1,\alpha]$. If M and m are respectively the maximum and the minimum values of $f$ in $[1,\alpha]$, then $3(M-m)$ is equal to :

For the function $f(x)=e^{\sin|x|}-|x|$, $x\in\mathbf R$, consider the following statements :

Statement I : $f$ is differentiable for all $x\in\mathbf R$.

Statement II : $f$ is increasing in $\left(-\pi,-\dfrac\pi2\right)$.

In the light of the above statements, choose the correct answer from the options given below :

Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f : \mathbb{R} \to \mathbb{R}$ by

$$f(x) = \begin{cases} 2 - 2x^2 - x^2\sin\dfrac{1}{x} & \text{if } x \neq 0,\\ 2 & \text{if } x = 0. \end{cases}$$

Then which one of the following statements is TRUE?

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.

Match each entry in List-I to the correct entry in List-II and choose the correct option.

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ be defined by

$$f(x) = \begin{cases} \dfrac{6x + \sin x}{2x + \sin x} & \text{if } x \neq 0,\\[2mm] \dfrac73 & \text{if } x = 0. \end{cases}$$

Then which of the following statements is (are) TRUE?

Let the function $f : \mathbb{R} \to \mathbb{R}$ be defined by

$$f(x) = \frac{\sin x}{e^{\pi x}}\cdot\frac{\left(x^{2023} + 2024x + 2025\right)}{\left(x^2 - x + 3\right)} + \frac{2}{e^{\pi x}}\cdot\frac{\left(x^{2023} + 2024x + 2025\right)}{\left(x^2 - x + 3\right)}.$$

Then the number of solutions of $f(x) = 0$ in $\mathbb{R}$ is _____.

Let $f : (0,1) \to \mathbb{R}$ be the function defined as $f(x) = [4x]\left(x - \dfrac14\right)^2\left(x - \dfrac12\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?

Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\dfrac{d^2f}{dx^2}(x) > 0$ for all $x \in (-1,1)$. For $f \in S$, let $X_f$ be the number of points $x \in (-1,1)$ for which $f(x) = x$. Then which of the following statements is(are) true?

For $x \in \mathbb{R}$, let $y(x)$ be a solution of the differential equation $(x^2-5)\dfrac{dy}{dx} - 2xy = -2x(x^2-5)^2$ such that $y(2) = 7$. Then the maximum value of the function $y(x)$ is _____.

Let
$$\alpha = \sum_{k=1}^{\infty} \sin^{2k}\left(\frac\pi6\right).$$
Let $g : [0, 1] \to \mathbb{R}$ be the function defined by
$$g(x) = 2^{\alpha x} + 2^{\alpha(1-x)}.$$
Then, which of the following statements is/are TRUE?

For $x \in \mathbb{R}$, let the function $y(x)$ be the solution of the differential equation
$$\frac{dy}{dx} + 12y = \cos\left(\frac{\pi}{12}x\right), \qquad y(0) = 0.$$
Then, which of the following statements is/are TRUE?

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

Let $x^2+y^2+Ax+By+C=0$ be a circle passing through $(0,6)$ and touching the parabola $y=x^2$ at $(2,4)$. Then $A+C$ is equal to ______.