Application of Derivatives — Previous Year Questions (Class 12 Mathematics)
Applications of Derivatives covers rate of change, increasing/decreasing functions, maxima/minima, and curve sketching. These concepts are essential for op
TL;DR: Applications of Derivatives covers rate of change, increasing/decreasing functions, maxima/minima, and curve sketching. These concepts are essential f…
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Applications of Derivatives covers rate of change, increasing/decreasing functions, maxima/minima, and curve sketching. These concepts are essential for op
Application of Derivatives — Previous Year Questions with Solutions
Q (2023, 3 marks): Find the intervals where f(x) = x³ - 3x² - 9x + 5 is increasing and decreasing.
Answer: Step 1: Find f'(x)
f'(x) = 3x² - 6x - 9
Step 2: Find critical points (where f'(x) = 0)
3x² - 6x - 9 = 0
x² - 2x - 3 = 0
(x - 3)(x + 1) = 0
x = 3 or x = -1
Step 3: Analyze sign of f'(x) in intervals (-∞, -1), (-1, 3), (3, ∞)
Test x = -2: f'(-2) = 3(4) - 6(-2) - 9 = 12 + 12 - 9 = 15 > 0
Test x = 0: f'(0) = -9 < 0
Test x = 4: f'(4) = 3(16) - 6(4) - 9 = 48 - 24 - 9 = 15 > 0
Increasing on: (-∞, -1) ∪ (3, ∞)
Decreasing on: (-1, 3)
Q (2022, 3 marks): Find the local maxima and minima of f(x) = x³ - 3x.
Answer: Step 1: Find f'(x)
f'(x) = 3x² - 3 = 3(x² - 1) = 3(x-1)(x+1)
Step 2: Critical points
3(x-1)(x+1) = 0
x = 1 or x = -1
Step 3: Second derivative test
f''(x) = 6x
At x = -1: f''(-1) = -6 < 0 → Local maximum
f(-1) = (-1)³ - 3(-1) = -1 + 3 = 2
Local maximum: (-1, 2)
At x = 1: f''(1) = 6 > 0 → Local minimum
f(1) = 1³ - 3(1) = 1 - 3 = -2
Local minimum: (1, -2)
Q (2023, 5 marks): A rectangular box is to be made from a square piece of cardboard of side 24 cm by cutting squares from corners and folding up. Find the maximum volume.
Answer: Let x = side of square cut from each corner
After folding: length = 24 - 2x, width = 24 - 2x, height = x
Volume V(x) = (24-2x)(24-2x)(x) = (24-2x)²·x
Expanding: V(x) = (576 - 96x + 4x²)·x = 576x - 96x² + 4x³
Step 1: Find V'(x)
V'(x) = 576 - 192x + 12x²
Step 2: Set V'(x) = 0
12x² - 192x + 576 = 0
x² - 16x + 48 = 0
(x - 4)(x - 12) = 0
x = 4 or x = 12
Since x must be < 12 (half of side length) and positive: x = 4 or x = 12
x = 12 is not valid (gives zero dimensions), so x = 4
Step 3: Verify maximum
V''(x) = 24x - 192
V''(4) = 96 - 192 = -96 < 0 → Maximum
V(4) = (24-8)(24-8)(4) = 16 × 16 × 4 = 1024 cm³
Maximum volume = 1024 cm³
Q (2021, 3 marks): At what rate is the surface area of a sphere increasing if the radius is increasing at 2 cm/s when r = 5 cm?
Answer: Surface area of sphere: S = 4πr²
Given: dr/dt = 2 cm/s, r = 5 cm
Differentiate with respect to time:
dS/dt = 4π · 2r · dr/dt
dS/dt = 8πr · dr/dt
Substitute values:
dS/dt = 8π(5)(2)
dS/dt = 80π cm²/s
dS/dt ≈ 251.33 cm²/s
Q (2022, 2 marks): Find the equation of the tangent to the curve y = x² - 2x at the point where x = 3.
Answer: Step 1: Find the point
At x = 3: y = (3)² - 2(3) = 9 - 6 = 3
Point: (3, 3)
Step 2: Find slope of tangent
dy/dx = 2x - 2
At x = 3: dy/dx = 2(3) - 2 = 6 - 2 = 4
Slope m = 4
Step 3: Equation of tangent
Using point-slope form: y - y1 = m(x - x1)
y - 3 = 4(x - 3)
y - 3 = 4x - 12
y = 4x - 9
Q (2023, 2 marks): The cost function for manufacturing x items is C(x) = 5x² + 20x + 50. Find the marginal cost when x = 10.
Answer: Marginal cost = dC/dx
C(x) = 5x² + 20x + 50
dC/dx = 10x + 20
At x = 10:
Marginal cost = 10(10) + 20 = 100 + 20 = 120
The marginal cost at x = 10 is 120 (per item)
Frequently Asked Questions
What is the difference between absolute and local extrema?
Local (relative) extrema are maximum/minimum values in a small neighborhood around a point. Absolute extrema are the largest/smallest values over the entire domain. An absolute extremum at an interior point is also a local extremum.
How do you use the first derivative test vs. second derivative test?
First derivative test: Examine sign change of f'(x) at critical point — if positive to negative, it's maximum; negative to positive, it's minimum. Second derivative test: Evaluate f''(x) at critical point — if f''(x) > 0, minimum; if f''(x) < 0, maximum; if f''(x) = 0, test is inconclusive.
More Class 12 Mathematics PYQs
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