Application of Derivatives — Class 12 solved problems

26 problems from this chapter, each solved step by step.

An empty inverted right circular conical vessel of base radius 12 cm and vertical height 18 cm is being filled with water at a constant rate of 12π cm³/s.At the
Total time elapsed is (55)/(3) s (or 18(1)/(3) s), and the water level rises instantly by ( [3]459 - 6) cm ≈ 1.71 cm.
Find the local maximum and minimum values of f(x) = x * sqrt(4 - x²).
The local maximum value is 2 at x = √2 and the local minimum value is -2 at x = - √2.
Find the maximum value of f(x) = 2x³ - 9x² + 12x + 5 on the interval [0, 3].
The maximum value of f(x) on the interval [0, 3] is 14.
Find the equation of the tangent and normal to the curve y = x³ - 3x + 2 at x = 1.
The equation of the tangent is y = 0. The equation of the normal is x = 1.
Find the local maxima and local minima of the function f(x) = sin(x) + cos(x) on the interval [0, 2*pi].
The function has a local maximum at x = (π)/(4) with value √2, and a local minimum at x = (5π)/(4) with value -√2.
The rate of change of the area of a circle with respect to its radius r (in cm²/s), when r = 6 cm is:
12π cm²/cm
(a) Find the intervals in R for which the function f(x) = x^4 - 2x² is increasing or decreasing. OR (b) Find: ∫ (2x + 1)/(√(18 - 4x - x²)) dx.
The function f(x) = x^4 - 2x² is increasing on the intervals (-1, 0) (1, ∞) and decreasing on the intervals (-∞, -1) (0, 1).
Let f(x)=∫_0^x t(t²-9t+20) dt, 1 ≤ x ≤ 5. If the range of f is [α, β], then 4(α+β) equals:
157
Let f: R → R be a polynomial function of degree four having extreme values at x = 4 and x = 5. If lim_x → 0 (f(x))/(x²) = 5, then f(2) is equal to:
10
A particle moves in the plane with position vector r⃗(t) = (t², e^t). Find its velocity, acceleration, and the tangential and normal components of acceleration
At t=1: Velocity v(1) = (2, e), Acceleration a(1) = (2, e), Tangential component of acceleration a_T = √(4 + e²), Normal component of acceleration a_N = 0.
Find the extrema of f(x,y,z) = xyz subject to the constraint x² + 2y² + 3z² = 6 using Lagrange multipliers.
The maximum value is (2√3)/(3) and the minimum value is -(2√3)/(3).
If the set of all values of a, for which the equation 5x³ - 15x - a = 0 has three distinct real roots, is the interval (α, β), then β - 2α is equal to
30
The sum of all local minimum values of the function f(x) = 1 - 2x, & x < -1 (1)/(3)(7 + 2|x|), & -1 ≤ x ≤ 2 (11)/(18)(x - 4)(x - 5), & x > 2 is
(157)/(72)
A sphere of radius r is inscribed in a cone with base radius R and height h. Find the relationship between r, R, and h.
The relationship between r, R, and h is r = (Rh)/(R+h).
Let x = -1 and x = 2 be the critical points of the function f(x) = x³ + ax² + b log_2|x| + 1, x 0. Let m and M respectively be the absolute minimum and the abso
23.68
Let the function f(x) = (x)/(3) + (3)/(x) + 3, x 0 be strictly increasing in (-∞, _1) ( _2, ∞) and strictly decreasing in ( _1, _2) ( _4, _5). Then Σ_i=1^(5) _i
18
Let f: [1, ∞) → [2, ∞) be a differentiable function. If ∫_1^x f(t) dt = 5x f(x) - x^5 - 9 for all x ≥ 1, then the value of f(3) is:
(274)/(27)
Let f(x)=∫_0^x² (t²-8t+15)/(e^t) dt, x R. Then the numbers of local maximum and local minimum points of f, respectively, are:
The number of local maximum points is 2 and the number of local minimum points is 3.
Let A(4, -2), B(1, 1) and C(9, -3) be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E, and F on the s
The maximum area of the parallelogram AFDE is 3 square units.
Find the maximum and minimum values of f(x) = sin(x) + cos(x) on the interval [0, π].
The maximum value of the function is √2 and the minimum value is -1 on the interval [0, π].
Let (2, 3) be the largest open interval in which the function f(x) = 2 log_e (x - 2) - x² + a x + 1 is strictly increasing and (b, c) be the largest open interv
360
If the function f(x)=2x³ - 9a x² + 12a² x + 1, where a>0, attains its local maximum and local minimum at p and q respectively, such that p² = q, then f(3) is eq
37
A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at th
256π cm²
EXTENSION: A sphere of radius r is inscribed in a cone with base radius R and height h. Find the relationship between r, R, and h. Additionally, generalize your
The relationship between r, R, and h is r = (Rh)/(R + √(R² + h²)). The generalization to n is not applicable as the problem describes a specific geometric confi
Consider the region R = (x, y): x ≤ y ≤ 9 - (11)/(3) x², x ≥ 0. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R,
The area of the largest rectangle is (567)/(121).
Find the equation of the tangent line to the curve y = x² - 2x + 7 which is parallel to the line 2x - y + 9 = 0.
The equation of the tangent line is y = 2x + 3.

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