Applications of Derivatives Questions - JEE

We have \[u^2 = a^2 + b^2 + 2\sqrt{(a^4+b^4)\sin^2\theta\cos^2\theta + a^2b^2(\sin^4\theta + \cos^4\theta)}\] The minimum value of $u^2$ is:

P is a variable point on the curve y = f(x) and A is a fixed point in the plane not lying on the curve. If PA is minimum, then the angle between PA and the tangent at P is

Let $g:\mathbb{R}\to\mathbb{R}$ be a non-constant twice differentiable function such that $g'\left(\dfrac{1}{2}\right)=g'\left(\dfrac{3}{2}\right)$. If a real valued function $f$ is defined as $f(x)=\dfrac{1}{2}[g(x)+g(2-x)]$, then

Let f(x) = \frac{x^2 + x - 1}{x^2 - x + 1}\, then the largest value of f(x)\ for x \in [-1, 3]\ is:

Let $f(x) = ax^2 - b|x|$, where $a$ and $b$ are constants. Then at $x = 0$, $f(x)$ has:

Let $f(x) = x^3 + x + 1$ and $g(x)$ be its inverse, then the equation of tangent to $y = g(x)$ at $x = 3$ is:

If $R_1$ = maximum range on a plane inclined up with inclination $\beta$ and $R_2$ = maximum range on a plane inclined down with inclination $\beta$, then which of the following is true?

The derivative of $f(x-3) \cdot g(x)$ with respect to x at $x = 100$ is

Let $f : \mathbb{R} \to \mathbb{R}$ be a differentiable function that satisfies the relation $f(x + y) = f(x) + f(y) - 1$, $\forall x, y \in \mathbb{R}$. If $f'(0) = 2$, then $|f(-2)|$ is equal to ____.

Let $f(x)=\begin{vmatrix}1+\sin^2x & \cos^2x & \sin2x\\\sin^2x & 1+\cos^2x & \sin2x\\\sin^2x & \cos^2x & 1+\sin2x\end{vmatrix}$, $x\in\left[\dfrac{\pi}{6},\dfrac{\pi}{3}\right]$. If $\alpha$ and $\beta$ are the maximum and minimum values of f, then:

The tangent at a point P on the curve $y = \ln\left(\frac{2 + \sqrt{4 - x^2}}{2 - \sqrt{4 - x^2}}\right) - \sqrt{4 - x^2}$ meets the y-axis at T; then $PT^2$ equals to:

The number of integral values of $a$ for which $f(x) = x^3 + (a+2)x^2 + 3ax + 5$ is monotonic on all $x \in \mathbb{R}$ is:

Given function is $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$. If $f(p) = \max. f(x)$ and $f(q) = \min. f(x)$, and $p^2 = q$, then the value of $a$ is:

Given that $g(x) = 2f\!\left(\dfrac{x^2}{2}\right) + f(6 - x^2)$, $\forall\, x \in R$ and $f''(x) > 0\; \forall\, x \in R$, then:

Let f(x) = x4 + ax3 + 3x2 + bx + 1, a, b ∈ ℝ. If f(x) ≥ 0 for all x ∈ ℝ, then the maximum value of a2 + b2 is equal to

The parametric form of a curve is $x = 2\cos t + 2t\sin t$ and $y = 2\sin t - 2t\cos t$. The distance from the origin of the normal to the curve at $t = \pi/4$ is:

Let $f(x) = \tan^{-1}\left(\frac{1-x}{1+x}\right)$. Then the difference of the greatest and least value of $f(x)$ on $[0,1]$ is:

If $|\ln x| = px$ has exactly three distinct solutions, then find $[p]$ (where $[.]$ denote greater integer function).

P and Q are two points on a circle of centre C and radius $a$, the angle PCQ being 20 then the radius of the circle inscribed in the triangle CPQ is maximum when

The minimum value of the function $f(x) = x^{3/2} + x^{-3/2} - 4\left(x + \dfrac{1}{x}\right)$ for all permissible real $x$, is:

Which of the following functions is increasing on $(-\infty, \infty)$?(1) $f(x) = x^3 - 3x^2 + 3x + 3$(2) $f(x) = 2x^3 - 3x^2 - 12x + 6$(3) $f(x) = 3x^2 - 2x + 1$

The minimum distance of a point on the curve $y = x^2 - 4$ from the origin is

Let $f$ and $g$ be twice differentiable functions on $\mathbb{R}$ such that $f''(x) = g''(x) + 6x$, $f'(1) = 4g'(1) - 3 = 9$, $f(2) = 3g(2) = 12$. Then which of the following is NOT true? (1) $g(-2) - f(-2) = 20$ (2) If $-1 < x < 2$, then $|f(x) - g(x)| < 8$ (3) $|f'(x) - g'(x)| < 6 \Rightarrow -1 < x < 1$ (4) There exists $x_0 \in \left(1, \frac{3}{2}\right)$ such that $f(x_0) = g(x_0)$

Given curve is $x^2 + 2xy - 3y^2 = 0$. The normal to the curve at $(1, 1)$ meets the curve again at which point?

If y = (x + sin x) + (x + sin x) + ..., then dy/dx = ________.

The number of points on the curve $y=54x^5-135x^4-70x^3+180x^2+210x$ at which the normal lines are parallel to $x+90y+2=0$ is:

If x = ∫₀ʸ f(t) dt and d²y/dx² = ay, then a is equal to

Let $f: [0, \infty) \to R$ be a continuous, strictly increasing function such that $f^3(x) = \int_0^x uf^2(u)du$. If a normal is drawn to the curve $y = f(x)$ with gradient $-\frac{1}{2}$, then find the intercept made by it on the $y$-axis.

Given line $y = x$ and curve $y^2 = x - 2$. The shortest distance between the line and the curve is:

Let $f(x)$ be a polynomial of degree four having extreme values at $x = 1$ and $x = 2$. If $\lim_{x \to 0}\left[1 + \dfrac{f(x)}{x^2}\right] = 3$, then $f(2)$ is equal to

The distance, from the origin, of the normal to the curve, $x = 2\cos t + 2t\sin t$, $y = 2\sin t - 2t\cos t$ at $t = \dfrac{\pi}{4}$, is

Let $f(x) = 1 + \int_0^1 (xe^y + ye^x)f(y)\,dy$ where $x$ and $y$ are independent variables.If the complete solution set of $x$ for which the function $h(x) = f(x) + 3x$ is strictly increasing is $(-\infty, k)$, then $e^k$ equals to: (where $[\cdot]$ denotes the greatest integer function)

Given that $\dfrac{dP}{dx} = 100 - 12\sqrt{x}$. The new level of production of items is $\displaystyle\int_{2000}^{P} dP = \int_{0}^{25} (100 - 12\sqrt{x})\,dx$. Find the value of $P$.

If $a > 0$, $b > 0$, $c > 0$ and $2a + b + 3c = 1$, then which of the following is correct?

The normal to the curve, $x^2 + 2xy - 3y^2 = 0$ at $(1, 1)$

Which of the following statements is true?

A manufacturing company makes two types of television sets; one is black and white and the other is in coloured. The company has resources to make at most 300 sets a week. It takes ₹1800 to make a black and white set and ₹2700 to make a coloured set. The company can spend not more than ₹648000 a week to make television sets. If it makes a profit of ₹510 per black and white set and ₹675 per coloured set, how many sets of each type should be produced so that the company has maximum profit? Formulate this problem as a LPP given that the objective is to maximise the profit.

If f(x) = 3x² + 15x + 5, then the approximate value of f(3.02) is

$l_1$ and $l_2$ are lengths of side of two variable squares $S_1$ and $S_2$ respectively for $l_i = l_1^2 + l_2^2 + 6$ at $l_2 = 1$. If rate of change of area of $S_2$ with respect to area of $S_1$ is equal to $\frac{1}{8m}$, then $m = $.

Given curve $y = f(x) = x^3 - x^2 - 2x$ Given point of line segment $A(1, f(1))$ and $B(-1, f(-1))$. The tangent of the curve is parallel to the line segment $AB$. Then the value of $|6\alpha + 2\beta|$ is:

From a given solid cone of height $H$, another inverted cone is carved whose height is $h$, such that its volume is maximum, then the ratio $\dfrac{H}{h}$ is equal to:

The equation of tangents to the curve y = cos(x − y), −2π ≤ x ≤ 2π that are parallel to the line x − 2y = 0, is

A value of C for which the conclusion of the mean value theorem holds for the function f(x) = logex on the interval [1, 3] is

A curve passes through $(2, 0)$ and the slope of tangent at any point $(x, y)$ is $x^2 - 2x$ $\forall\, x \in R$. The point of minimum ordinate on the curve where $x > 0$ is $(a, b)$, then find the value of $(a + 6b)$.

From a point on the curve $y = x^2 - 4$, the minimum distance from the origin is:

Let f(x) = 4 tan x − tan² x + tan³ x, where x ≠ nπ + π/2, n ∈ ℤ, then

100. If $f(x) = x^3 - 3x + 1$, then minimum number of real roots of $f(f(x)) = 0$ is:

Twenty metres of wire is available for fencing off a flower-bed in the form of a circular sector. Then, the maximum area (in sq. m) of the flower-bed, is

Let f(x) = ∫₋ₘ² e^((1+t)²) dt and g(x) = f(h(x)), where h(x) is defined for all x ∈ ℝ. If g'(2) = e⁴ and h'(2) = 1, then the absolute value of the sum of all possible values of h(2) is ________.

If P(x) has real roots α, β, γ, then [α] + [β] + [γ] is

Applications of Derivatives Questions (514)