Monotonicity

Let the function $f(x)=2x^2-\log_ex$, $x>0$, be decreasing in $(0,a)$ and increasing in $(a,4)$. A tangent to the parabola $y^2=4ax$ at a point P on it passes through the point $(8a,8a-1)$ but does not pass through the point $\left(-\dfrac1a,0\right)$. If the equation of the normal at P is $\dfrac x\alpha+\dfrac y\beta=1$, then $\alpha+\beta$ is equal to ______.

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

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?

The function $f(x)=xe^{x(1-x)}$, $x\in\mathbf R$, 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 $x>\sqrt2$

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?

Which of the following statements is TRUE?

Which of the following statements is TRUE?

The function $f(x)=\dfrac{4x^3-3x^2}6-2\sin x+(2x-1)\cos x$ :

If the function $f:\mathbb R\to\mathbb R$ is defined by $f(x) = |x|(x - \sin x)$, then which of the following statements is TRUE?

Let $b$ be a nonzero real number. Suppose $f:\mathbb R\to\mathbb R$ is a differentiable function such that $f(0) = 1$. If the derivative $f'$ of $f$ satisfies the equation

$$f'(x) = \frac{f(x)}{b^2 + x^2}$$

for all $x\in\mathbb R$, then which of the following statements is/are TRUE?

Let $f:\mathbb R\to\mathbb R$ be given by

$$f(x) = \begin{cases}x^5 + 5x^4 + 10x^3 + 10x^2 + 3x + 1, & x < 0;\\ x^2 - x + 1, & 0 \le x < 1;\\ \frac23x^3 - 4x^2 + 7x - \frac83, & 1 \le x < 3;\\ (x-2)\log_e(x-2) - x + \frac{10}3, & x \ge 3.\end{cases}$$

Then which of the following options is/are correct?

Let

$$f(x) = \frac{\sin\pi x}{x^2},\quad x > 0.$$

Let $x_1 < x_2 < x_3 < \cdots < x_n < \cdots$ be all the points of local maximum of $f$ and $y_1 < y_2 < y_3 < \cdots < y_n < \cdots$ be all the points of local minimum of $f$. Then which of the following options is/are correct?

For every twice differentiable function $f: \mathbb{R} \to [-2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$, which of the following statement(s) is (are) TRUE?

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be two non-constant differentiable functions. If

$$f'(x) = \left(e^{(f(x) - g(x))}\right) g'(x) \quad \text{for all } x \in \mathbb{R},$$

and $f(1) = g(2) = 1$, then which of the following statement(s) is (are) TRUE?

Which of the following options is the only CORRECT combination?

Which of the following options is the only CORRECT combination?

Which of the following options is the only INCORRECT combination?

If $f: \mathbb{R} \to \mathbb{R}$ is a twice differentiable function such that $f''(x) > 0$ for all $x \in \mathbb{R}$, and $f\left(\frac{1}{2}\right) = \frac{1}{2}$, $f(1) = 1$, then

If $f: \mathbb{R} \to \mathbb{R}$ is a differentiable function such that $f'(x) > 2f(x)$ for all $x \in \mathbb{R}$, and $f(0) = 1$, then