Is it correct to say: Confusing local extremum with global extremum?

Global extremum on [a,b] requires checking endpoints too. Local extremum is only about neighbourhood.

Is it correct to say: Applying second-derivative test blindly when f''(x₀) = 0?

If f''(x₀) = 0, second test is inconclusive — use first-derivative test instead.

Is it correct to say: Forgetting to check conditions of Rolle's / MVT before applying?

State f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b) for Rolle's. Missing condition = mark cut.

Is it correct to say: Using dy/dx = 0 alone to conclude extremum without sign analysis?

dy/dx = 0 is necessary but not sufficient. Could be an inflexion point (e.g. y = x³ at x = 0).

Is it correct to say: Confusing tangent slope with normal slope?

m_normal = −1/m_tangent, not the negative slope.

Is it correct to say: Ignoring units in rate-of-change problems?

dV/dt is in cm³/s if V is in cm³ and t in s. Board asks for units.

Applications of Derivatives

Rate of change, increasing/decreasing, tangents, normals, maxima, minima — Maharashtra HSC Maths Ch 4

📐 12 formulas✏️ 3 examples🎯 5 practice⚖️ 6 marks🏫 Maharashtra State Board📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Maxima/minima problem statement (open box from a rectangular sheet, cylinder inscribed in a sphere) is a 4-mark HSC classic.
→Approximate value using dy = f'(x)dx (e.g. √25.3 ≈ 5.03) is asked every year for 2 marks.
→Rate of change (dV/dt for expanding balloon, water flow into cone) is a 3-marker.
→Rolle's theorem and Lagrange's Mean Value Theorem: state, verify conditions, find c ∈ (a,b) — Maharashtra Board favourite 3-mark question.
→Draw a sign chart of f'(x) to establish where the function is increasing/decreasing.

📐 Formulas(12)

Rate of Change★ Board fav

Rate = \frac{d y}{d t} = \frac{d y}{d x} \cdot \frac{d x}{d t}

Approximation (Differential)★ Board fav

f (x + Δ x) \approx f (x) + f^{'} (x) Δ x

Slope of Tangent and Normal

m_{t an} = f^{'} (x_{0}), m_{n or ma l} = - \frac{1}{f ^{'} ( x _{0} )}

Equation of Tangent Line

y - y_{0} = f^{'} (x_{0}) (x - x_{0})

Increasing / Decreasing on Interval★ Board fav

f^{'} (x) > 0 \Rightarrow increasing; f^{'} (x) < 0 \Rightarrow decreasing

First Derivative Test (extrema)★ Board fav

f^{'} (x_{0}) = 0, f^{'} (x) changes sign at x_{0}

Second Derivative Test★ Board fav

f^{'} (x_{0}) = 0, f^{''} (x_{0}) < 0 \Rightarrow local max; f^{''} (x_{0}) > 0 \Rightarrow local min

Global Maximum on Closed Interval

max_{[a, b]} f = max {f (a), f (b), f (c_{i})}

Rolle's Theorem★ Board fav

f continuous [a, b], differentiable (a, b), f (a) = f (b) \Rightarrow \exists c \in (a, b) : f^{'} (c) = 0

Lagrange's Mean Value Theorem

\exists c \in (a, b) : f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}

Percentage Error

% error = \frac{Δ y}{y} \times 100 \approx \frac{f ^{'} ( x ) Δ x}{f ( x )} \times 100

Angle of Intersection of Curves

tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}

✏️ Solved Examples

1Solved Exampleeasy3 steps

A spherical balloon is being inflated. Find the rate of increase of its volume when the radius is 5 cm and dr/dt = 0.5 cm/s.

Volume of sphere.

V = \frac{4}{3} π r^{3}

2Solved Exampleboard4 steps

Find the maximum and minimum values of f(x) = x³ − 6x² + 9x + 15 on [0, 6].

Compute f'(x).

f^{'} (x) = 3 x^{2} - 12 x + 9 = 3 (x - 1) (x - 3)

3Solved ExampleHOTS5 steps

A rectangular box open at the top is made from a rectangular sheet of side 24 cm × 15 cm by cutting equal squares of side x from each corner and folding. Find the value of x for which the volume is maximum.

Volume of box.

V (x) = x (24 - 2 x) (15 - 2 x)

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Confusing local extremum with global extremum
✓Global extremum on [a,b] requires checking endpoints too. Local extremum is only about neighbourhood.
2
Applying second-derivative test blindly when f''(x₀) = 0
✓If f''(x₀) = 0, second test is inconclusive — use first-derivative test instead.
3
Forgetting to check conditions of Rolle's / MVT before applying
✓State f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b) for Rolle's. Missing condition = mark cut.
4
Using dy/dx = 0 alone to conclude extremum without sign analysis
✓dy/dx = 0 is necessary but not sufficient. Could be an inflexion point (e.g. y = x³ at x = 0).
5
Confusing tangent slope with normal slope
✓m_normal = −1/m_tangent, not the negative slope.
6
Ignoring units in rate-of-change problems
✓dV/dt is in cm³/s if V is in cm³ and t in s. Board asks for units.

🎯 Practice Yourself

🎯Practice Yourself5 questions

Q1
The volume of a cube is increasing at 8 cm³/s. Find rate of change of surface area when edge = 4 cm.
Q2
Find the equation of the tangent to y = x² − 3x + 2 at x = 1.
Q3
Verify Rolle's theorem for f(x) = sin x on [0, π].
Q4
Find the intervals on which f(x) = x³ − 3x² + 4 is increasing.
Q5
Approximate √50.

📝 Notes

Applications of Derivatives — Maharashtra HSC Overview

Chapter 4 of the Balbharati Class 12 Mathematics textbook is application-heavy: rate of change, tangents/normals, monotonicity, maxima/minima, approximations, and mean-value theorems.

Maharashtra syllabus specifics

Rolle's theorem and Lagrange's MVT — formal proof and verification questions are Maharashtra-specific. CBSE lists them but rarely asks proofs.
Word problems on maxima/minima — open box, sector of circle, rectangle inscribed in circle, cylinder in sphere, most economical container — are a Balbharati staple.
Approximation using differentials for common calculations (√25.3, ³√126, sin 30° 1′) is a Maharashtra HSC 2-mark favourite.
Angle between two curves (using tan θ = |(m₁ − m₂)/(1 + m₁m₂)|) is asked more often in Maharashtra than CBSE.

Strategy

Draw a diagram for every word problem — it earns 1 mark by itself.
For maxima/minima: identify the variable, express the target function in one variable, differentiate, set to zero, verify with second-derivative test.
State conditions of MVT before applying — 1 mark per condition stated.
For rate-of-change problems, identify the CHAIN: dy/dt = (dy/dx)(dx/dt).