Applications of Derivatives Questions (514)

If non-zero real numbers \(b\) and \(c\) are such that \(\min f(x) > \max g(x)\), where \(f(x) = x^2 + 2bx + 2c^2\) and \(g(x) = -x^2 - 2cx + b^2\) (\(x \in \mathbb{R}\)); then \(\left|\dfrac{c}{b}\right|\) lies in the interval:

A spherical balloon is being inflated at a constant rate. The volume of the balloon after 49 minutes of leakage is \(4500\pi - 49(72\pi) = 972\pi\). The rate (in meters per minute) at which the radius of the balloon decreases 49 min after the leakage began is:

A function \(y = f(x)\) has a second order derivative \(f''(x) = 6(x-1)\). If its graph passes through the point (2, 1) and at that point the tangent to the graph is \(y = 3x - 5\), then the function is

\(P_r(x_r, y_r);\, r = 1, 2, 3, \ldots\) is a sequence of points on the curve \(y^3 = x\). The tangent at \(P_n\) cuts the curve again at \(P_{n+1}\) for all \(n \in \mathbb{N}\). Then \(y_1, y_2, y_3, \ldots\) are in

The sum of the maximum and minimum values of the function f(x) = \sin\left(\frac{1}{2x}\right) + \cos\left(\frac{1}{2x}\right) + \sec\left(\frac{1}{2x}\right) is

Let f(x) = x cos−1(−sin|x|), x ∈ [−π/2, π/2]. Which of the following is true?

Given that \(x\), \(y\), \(z\) are positive reals such that \(xyz = 32\). The minimum value of \(x^2 + 4xy + 4y^2 + 2z^2\) is ______.

Given f(x) = x√(kx − x²). If f(x) is an increasing function on [0, 3] and M is the maximum value of f(x), find the minimum value of k (denoted m) such that f is increasing on [0,3], and find M. What is the value of M (approximately)?

Let \(C\) be a curve given by \(y(x) = 1 + \sqrt{4x - 3}\), \(x > 3/4\). If \(P\) is a point on \(C\), such that the tangent at \(P\) has slope \(2/3\), then a point through which the normal at \(P\) passes, is

Let f and g be two differentiable functions on R such that f'(x) > 0 and g'(x) < 0, for all x ∈ R. Then for all x:

Find the maximum and minimum values of \[ f(x) = x(1-x)^2, \quad 0 \leq x \leq 2. \]

If \(f(x)\) is continuous and derivable on \([-2, 5]\) and \(-4 \leq f'(x) \leq 3\) \(\forall\, x \in (-2, 5)\), then difference of maximum and minimum value of \(f(5)\) is equal to:

Let \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) be a polynomial such that \( P'(x) = 4x^3 + 3ax^2 + 2bx + c \). Given that \( P(-1)

Find the vertical angle of a right circular cone of minimum curved surface area that circumscribes a given sphere.

If \(y = Ae^{mx} + Be^{nx}\), then the value of \(\frac{d^2y}{dx^2} - (m+n)\frac{dy}{dx} + mny\) is equal to

For the function \(y = f(x) = (x - b)x(x - c)e^x\), which of the following holds?

The equation of a normal to the curve, \(\sin y = x\sin\left(\dfrac{\pi}{3} + y\right)\) at \(x = 0\), is

If the function \(f(x) = (x^2 + ax + 2a)e^x\) is a strictly increasing function in \((-\infty, \infty)\). Then the number of integral values of \(a\) is:

If \(y + x + y = x = c\), where \(c > 0\), then \(\frac{dy}{dx}\) has the value equal to

Consider \( f(x) = \tan^{-1}\!\left(\sqrt{\dfrac{1+\sin x}{1-\sin x}}\right), x \in \left(0, \dfrac{\pi}{2}\right) \). A normal to \( y = f(x) \) at \( x = \dfrac{\pi}{6} \) also passes through the point

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

Given quadratic equation is \(ax^2 + bx + c = 0\) Let \(f'(x) = ax^2 + bx + c\), so \(f(x) = \frac{ax^3}{3} + \frac{bx^2}{2} + cx\). Clearly, \(f(0) = 0\) and \(f(1) = \frac{1}{6}(2a + 3b + 6c) = 0\). Given that \(f(0) = 0 = f(1)\), by Rolle's theorem, \(f'(x)\) has at least one root in \((0, 1)\). The root of \(ax^2 + bx + c = 0\) lies in:

Let \(f(x) = \begin{cases} \frac{1}{2}x^3 - x^2 + 10x + 5, & x \leq 1 \\ 4 - 2^x + \log_2(b^2 - 2), & x > 1 \end{cases}\)The set of values of \(b\) for which \(f(x)\) has greatest value at \(x = 1\) is given by

49. Consider the function \(f(x) = |x^2 - 7x + 12|(x^2 - 7x + 10)(x^2 - 4x + 3)\). Then Rolle's theorem for \(f(x)\) is not applicable to which of the following range?

If \(x\) and \(y\) are given parametrically, then \(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt} = \dfrac{\sin t}{\cos t}\). The length of the tangent is:

The absolute minimum and maximum values of the function \(\frac{x^2 - x + 1}{x^2 + x + 1}\) are given by

66. (A) In a circle, \(r = 5\) cm and \(\Delta r = 0.06\) cm. Find the approximate change in area \(\Delta A\) using differentials, where \(A = \pi r^2\).

Given functions are f(x) = 5 - |x - 2| and g(x) = |x + 1|, where x \in \mathbb{R}. Find ab where a is the value of x at which maximum of f(x) occurs and b is the value of x at which minimum of g(x) occurs.

Let \(y = \sqrt{x + \sqrt{x + \sqrt{x + \cdots}}}\), then \(\frac{dy}{dx}\) equals

If \(2^x + 2^y = 2^{x+y}\), then \(\frac{dy}{dx}\) is equal to

The volume of a right circular cone is given by \(V = \frac{1}{3}\pi r^2 h\). When \(h = 20\) cm, \(\frac{dh}{dt} = -4\) cm/s, \(r = 10\) cm and \(\frac{dr}{dt} = 2\) cm/s, find \(\frac{dV}{dt}\) (in cm³/s).

The minimum value of the function f(x) = x^{3/2} - x^{-3/2} + 4 for all permissible real values of x, is

The set of values of \(p\) for which \(f(x) = p^2x - \int 2^{4-x^2}\,dx\) is increasing for all \(x \in R\), is:

Let f(x) = x² − 2x and g(x) = f(f(x) − 1) + f(5 − f(x)), then

Let \(f(x)\) be a differentiable function on \([0, 8]\) such that \(f(1) = 3\), \(f(2) = 1/2\), \(f(3) = 4\), \(f(4) = -2\), \(f(5) = 6\), \(f(6) = 1/3\), \(f(7) = -1/4\). Then the minimum number of points of interaction of the curve \(y = f'(x)f(x)^2\) and \(y = f'(x)f(x)^2\) is \(k\), then find \(k\).

The normal to the curve \(y(x-2)(x-3) = x + 6\) at the point where the curve intersects the y-axis passes through the point:

Suppose, \(f(x) = e^{ax} + e^{bx}\), where \(a \neq b\) and \(f''(x) - 2f'(x) - 15f(x) = 0\) for all \(x \in \mathbb{R}\). Then, find \(ab\).

Which of the following could be the sketch of graph of \(y = \frac{d}{dx}(x \log x)\)?

For any \(x, y \in R\), \(xy > 0\). Then the minimum value of \(\dfrac{2x}{y^3} + \dfrac{x^3 y}{3} + \dfrac{4y^2}{9x^4}\) is ______.

Given \(f'(x) = x^2 - 2x\) and \(f(2) = 0\), the curve \(y = f(x)\) has a minimum at \(x = 2\). If the point of minimum ordinate is \((a, b)\), find \(a + 6b\).

On [1, e], the least and greatest values of f(x) = x2 ln x are m and M respectively, then [M + m] is: (where [ ] denotes greatest integer function)

For \(x \in \left(0, \dfrac{5\pi}{2}\right)\), define \(f(x) = \int_0^x \sqrt{t} \sin t \, dt\). Then \(f\) has

Given \(f(x) = x^2 + \dfrac{1}{x^2}\) and \(g(x) = x - \dfrac{1}{x}\),\(h(x) = \dfrac{f(x)}{g(x)}\). The local maximum and local minimum values of \(h(x)\) are respectively:

Let \( f(x) = \dfrac{3x^2 + 9x + 17}{3x^2 + 9x + 7} \) Find the maximum value of \( f(x) \).

Let \(f: \mathbb{R} \to \mathbb{R}\) be defined by \(f(x) = \begin{cases} k - 2x, & \text{if } x \leq -1 \\ 2x + 3, & \text{if } x > -1 \end{cases}\). If \(f\) has a local minimum at \(x = -1\), then a possible value of \(k\) is

Let \(f(x) = 30 - 2x - x^3\). If \(f(f(f(x))) > f(f(-x))\), find the highest integral value of \(x\) satisfying this inequality.

If \(y = \frac{\sec x + \tan x}{\sec x - \tan x}\), then \(\frac{dy}{dx}\) equals

The value of \(a\) for which the function \(f(x) = (4a-3)(x + \log 5) + 2(a-7)\cot\dfrac{x}{2}\sin^2\dfrac{x}{2}\) does not possess critical points is

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side x units and a circle of radius r units. If the sum of the areas of the square and the circle so formed is minimum, then:

Ex. 69 The function \(f(x)\) for \(-a \leq x \leq a\) is