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Application of Derivatives – Class 12 Mathematics notes, important questions and MCQs

NCERT Class 12 Mathematics, chapter 6. Notes, key terms, important questions with answers and MCQs.

Chapter overview

Derivatives measure rates of change, find tangents, tell where functions increase or decrease, and locate maxima and minima.

Rates and monotonicity

Rate of change: dA/dt = (dA/dr)(dr/dt) for related rates.
f′(x) > 0 ⇒ increasing; f′(x) < 0 ⇒ decreasing.
Slope of tangent at (x₁, y₁) = f′(x₁); normal slope = −1/f′(x₁).

Maxima and minima

Critical points: f′(x) = 0 or undefined.
Second derivative test: f″(c) < 0 ⇒ local maximum; f″(c) > 0 ⇒ local minimum.
Absolute extrema on [a, b]: compare values at critical points and endpoints.

Key terms

Critical point: point where f′(x) = 0 or does not exist
Local maximum: point where f is larger than at nearby points
Point of inflection: a point where the curve changes concavity; f″ changes sign there

Formulas

Tangent: y − y₁ = f′(x₁)(x − x₁)
f″(c) < 0 ⇒ max; f″(c) > 0 ⇒ min

Important questions with answers

Find the rate of change of the volume of a sphere with respect to its radius when r = 3 cm.2 marks

dV/dr = 4πr² = 36π cm³/cm.

Find the intervals in which f(x) = 2x³ − 9x² + 12x + 15 is increasing or decreasing.3 marks

f′(x) = 6x² − 18x + 12 = 6(x − 1)(x − 2). Increasing on (−∞, 1) and (2, ∞); decreasing on (1, 2).

An open box is made from a square sheet of side 18 cm by cutting equal squares from the corners and folding up. Find the side of the squares to be cut for maximum volume and the maximum volume.5 marks

Let side cut = x. V = x(18 − 2x)², 0 < x < 9. dV/dx = (18 − 2x)² − 4x(18 − 2x) = (18 − 2x)(18 − 6x). dV/dx = 0 gives x = 3 (x = 9 gives zero volume). d²V/dx² = 24x − 144 = −72 < 0 at x = 3, so maximum. V = 3 × 12² = 432 cm³.

Quick revision

f′ > 0 means increasing.
Second derivative test decides max/min.
Check endpoints for absolute extrema.

Exam tips

In optimisation word problems, express the quantity in one variable before differentiating.
In word problems, always check the second derivative or the endpoints to confirm max or min.

Practice MCQs

If f′(x) > 0 on an interval, f is
1. decreasing
2. increasing
3. constant
4. zero
Show answer
Answer: (b) increasing. Positive slope.
Slope of the tangent to y = x² at x = 3 is
1. 3
2. 9
3. 6
4. 2
Show answer
Answer: (c) 6. 2x.
The radius of a circle increases at 2 cm/s. Rate of increase of area when r = 5 cm is
1. 10π cm²/s
2. 4π cm²/s
3. 25π cm²/s
4. 20π cm²/s
Show answer
Answer: (d) 20π cm²/s. 2πr × 2.
f(x) = x³ − 3x has a local maximum at
1. x = 1
2. x = 3
3. x = 0
4. x = −1
Show answer
Answer: (d) x = −1. f″(−1) = −6 < 0.
f(x) = x³ − 6x² + 9x + 15 is decreasing on
1. (−∞, 1)
2. (1, 3)
3. (3, ∞)
4. (0, 1)
Show answer
Answer: (b) (1, 3). f′ = 3(x − 1)(x − 3) < 0.
Maximum value of sin x + cos x is
1. 1
2. 2
3. √2
4. 1/√2
Show answer
Answer: (c) √2. √2 sin(x + π/4).
Two positive numbers with sum 20 have maximum product when they are
1. 15 and 5
2. 19 and 1
3. 12 and 8
4. 10 and 10
Show answer
Answer: (d) 10 and 10. Product x(20 − x) max at x = 10.
The equation of the normal to y = x² at (1, 1) is
1. y = 2x − 1
2. 2x + y = 3
3. x + 2y = 3
4. x − 2y = −1
Show answer
Answer: (c) x + 2y = 3. Slope of normal −1/2.

All chapters of Class 12 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.