Applications of Derivatives

The value of $6m_2 + 4n_2 + 8m_2n_2$ is ____.

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 tangent to the curve $y=x^3$ at the point P(t, t$^3$) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 : 2 is :

If the least and the largest real values of α, for which the equation $z+\alpha|z-1|+2i=0$ ($z\in\mathbb C$ and $i=\sqrt{-1}$) has a solution, are p and q respectively; then $4(p^2+q^2)$ is equal to _______.

The minimum value of α for which the equation $\dfrac4{\sin x}+\dfrac1{1-\sin x}=\alpha$ has at least one solution in $\left(0,\dfrac\pi2\right)$ is _______.

An inclined plane is bent in such a way that the vertical cross-section is given by $y=\dfrac{x^2}4$ where y is in vertical and x in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction μ = 0.5, the maximum height in cm at which a stationary block will not slip downward is _______ cm.

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?

Consider all rectangles lying in the region

$$\left\{(x,y)\in\mathbb R\times\mathbb R : 0 \le x \le \frac\pi2 \text{ and } 0 \le y \le 2\sin(2x)\right\}$$

and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

For a polynomial $g(x)$ with real coefficients, let $m_g$ denote the number of distinct real roots of $g(x)$. Suppose $S$ is the set of polynomials with real coefficients defined by

$$S = \left\{(x^2-1)^2\left(a_0 + a_1x + a_2x^2 + a_3x^3\right) : a_0, a_1, a_2, a_3 \in \mathbb R\right\}.$$

For a polynomial $f$, let $f'$ and $f''$ denote its first and second order derivatives, respectively. Then the minimum possible value of $\left(m_{f'} + m_{f''}\right)$, where $f\in S$, is ______

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 the function $f:(0,\pi)\to\mathbb R$ be defined by

$$f(\theta) = (\sin\theta + \cos\theta)^2 + (\sin\theta - \cos\theta)^4.$$

Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta\in\{\lambda_1\pi, \ldots, \lambda_r\pi\}$, where $0 < \lambda_1 < \cdots < \lambda_r < 1$. Then the value of $\lambda_1 + \cdots + \lambda_r$ is ______

Define the collections $\{E_1, E_2, E_3, \ldots\}$ of ellipses and $\{R_1, R_2, R_3, \ldots\}$ of rectangles as follows:

$E_1: \dfrac{x^2}9 + \dfrac{y^2}4 = 1$;

$R_1$: rectangle of largest area, with sides parallel to the axes, inscribed in $E_1$;

$E_n$: ellipse $\dfrac{x^2}{a_n^2} + \dfrac{y^2}{b_n^2} = 1$ of largest area inscribed in $R_{n-1}$, $n > 1$;

$R_n$: rectangle of largest area, with sides parallel to the axes, inscribed in $E_n$, $n > 1$.

Then which of the following options is/are correct?

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 $\Gamma$ denote a curve $y = y(x)$ which is in the first quadrant and let the point $(1, 0)$ lie on it. Let the tangent to $\Gamma$ at a point $P$ intersect the $y$-axis at $Y_P$. If $PY_P$ has length 1 for each point $P$ on $\Gamma$, 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?

Let $f:\mathbb R\to\mathbb R$ be given by $f(x) = (x-1)(x-2)(x-5)$. Define

$$F(x) = \int_0^xf(t)\,dt,\quad x > 0.$$

Then which of the following options is/are correct?

Let $\vec a = 2\hat i + \hat j - \hat k$ and $\vec b = \hat i + 2\hat j + \hat k$ be two vectors. Consider a vector $\vec c = \alpha\vec a + \beta\vec b$, $\alpha, \beta\in\mathbb R$. If the projection of $\vec c$ on the vector $(\vec a + \vec b)$ is $3\sqrt2$, then the minimum value of $(\vec c - (\vec a\times\vec b))\cdot\vec c$ equals ______

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?