Summary and Quick Revision

The Chapter in One Idea

Everything in this chapter is one statement wearing different costumes:

$f'(x)$ is the instantaneous rate of change of $f$ — and geometrically, the slope of the tangent.

From this: rates of change of physical quantities (Section 1), where a function rises and falls (Section 2), where its peaks and valleys sit (Section 3), and how to optimise real quantities (Section 4).

Exam weight at a glance: CBSE Boards — typically one 2-mark rate/marginal question, one 3-mark monotonicity or local-extrema question, and a 5-mark optimisation problem. JEE Main — 1-2 questions, almost always monotonicity (often with a parameter) or maxima-minima. JEE Advanced adds tangents/normals and MVT (see JEE Corner).

Rate of Change — Formula Card

• $\frac{dy}{dx}$ = rate of change of $y$ with respect to $x$; $\left.\frac{dy}{dx}\right|_{x=x_0}$ = rate at the instant $x = x_0$.
• $\frac{dy}{dx} > 0$: $y$ increases as $x$ increases; $\frac{dy}{dx} < 0$: $y$ decreases. A decreasing quantity enters equations with a negative rate.
• Related rates (Chain Rule): if $x = f(t)$, $y = g(t)$, then $\frac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$ (for $\frac{dx}{dt} \neq 0$). In practice: differentiate the connecting equation with respect to $t$, substitute values after differentiating.
• Marginal cost / revenue: MC $= \frac{dC}{dx}$, MR $= \frac{dR}{dx}$ — differentiate, then substitute the output level.

Rates you should be able to write blindfolded:

Quantity	Formula	Rate
Circle area	$A = \pi r^2$	$\frac{dA}{dt} = 2\pi r\frac{dr}{dt}$
Circumference	$C = 2\pi r$	$\frac{dC}{dt} = 2\pi\frac{dr}{dt}$
Sphere volume	$V = \frac{4}{3}\pi r^3$	$\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}$
Cube volume / surface	$V = x^3$, $S = 6x^2$	$3x^2\frac{dx}{dt}$, $12x\frac{dx}{dt}$
Cone volume	$V = \frac{1}{3}\pi r^2 h$	eliminate $r$ or $h$ via the given proportion first

Key Point: $\frac{dV}{dr}$ for a sphere equals its surface area $4\pi r^2$ — growth happens by "adding a skin".

Monotonicity — Formula Card

Definitions (on an interval $I$): increasing: $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$; decreasing: $x_1 < x_2 \Rightarrow f(x_1) > f(x_2)$; constant: $f(x) = c$.

Theorem 1 (the workhorse). $f$ continuous on $[a,b]$, differentiable on $(a,b)$:

• $f'(x) > 0$ on $(a,b)$ $\Rightarrow$ $f$ increasing on $[a,b]$
• $f'(x) < 0$ on $(a,b)$ $\Rightarrow$ $f$ decreasing on $[a,b]$
• $f'(x) = 0$ on $(a,b)$ $\Rightarrow$ $f$ constant

Working method: find $f'$, solve $f'(x) = 0$, split the domain, make a sign table, conclude interval-by-interval. Never union two increasing intervals into one claim.

Templates to remember:

• $f'$ a perfect square (e.g. $3(x-1)^2 + 1$ or $6(x-1)^2$) $\Rightarrow$ monotonic despite $f' = 0$ at a point — isolated zeros of $f'$ don't break strict monotonicity ($x^3$ is the model).
• Parameter template: $px^2 + qx + r \geq 0\ \forall x \iff p > 0,\ q^2 - 4pr \leq 0$.
• Inequality template: to show $g > h$ on $(0, \infty)$: set $F = g - h$, check $F(0) = 0$ and $F' > 0$. Classics: $\frac{x}{1+x} < \log(1+x) < x$; $\sin x < x < \tan x$ on $\left(0, \frac{\pi}{2}\right)$; $e^x > 1 + x$.

Maxima and Minima — Formula Card

• Critical point: $f'(c) = 0$ or $f$ not differentiable at $c$ (corners count!). Only critical points can host local extrema (Theorem 2) — but not every critical point does ($x^3$ at 0).
• First Derivative Test: through the critical point $c$, $f'$ changes $(+) \to (-)$: local max; $(-) \to (+)$: local min; no change: point of inflexion.
• Second Derivative Test: $f'(c) = 0$ and $f''(c) < 0$: local max; $f''(c) > 0$: local min; $f''(c) = 0$: test fails — go back to the first derivative test.
• Absolute extrema on $[a, b]$ (closed-interval method): evaluate $f$ at all critical points in $(a,b)$ AND at both endpoints; the largest value is the absolute maximum, the smallest the absolute minimum. A continuous function on a closed interval always attains both (Theorem 5).
• Optimisation recipe: name variables (draw the figure) → write the target → use the constraint to reach ONE variable → state the domain → derivative test → answer the actual question with units.

Key Point: "Local maximum value" means $f(c)$, not $c$. Always report both the point and the value.

Standard Results and JEE Bounds — Quick Card

Optimisation punchlines (learn the setups in Sections 4-6):

• Fixed sum $k$: product maximised at $\frac{k}{2}, \frac{k}{2}$; sum of squares minimised at $\frac{k}{2}, \frac{k}{2}$; $x^m y^n$ maximised at $\frac{mk}{m+n}, \frac{nk}{m+n}$.
• Rectangle in a circle: square; rectangle of fixed perimeter: square.
• Open box from an $a \times a$ sheet: cut $x = \frac{a}{6}$.
• Cylinder of given surface, max volume: $h = 2r$. Cylinder in a sphere (radius $R$): $h = \frac{2R}{\sqrt 3}$.
• Cylinder in a cone: greatest curved surface at $x = \frac{r}{2}$; greatest volume at $x = \frac{2r}{3}$ (height $\frac{h}{3}$).
• Cone in a sphere: height $\frac{4r}{3}$, volume $\frac{8}{27}$ of the sphere.
• Cone of given slant height, max volume: $\alpha = \tan^{-1}\sqrt{2}$; given total surface, max volume: $\alpha = \sin^{-1}\frac{1}{3}$; given volume, least curved surface: $h = \sqrt{2}\,r$.
• Wire of length $L$ → square + circle, minimum total area: square piece $\frac{4L}{\pi+4}$, circle piece $\frac{\pi L}{\pi+4}$.

JEE bounds to quote:

• $a\sin x + b\cos x \in [-\sqrt{a^2+b^2}, \sqrt{a^2+b^2}]$
• $x + \frac{a^2}{x} \geq 2a$ ($x > 0$)
• $\frac{\log x}{x}$ peaks at $e$ (value $\frac{1}{e}$); $x^{1/x}$ peaks at $e$ (value $e^{1/e}$); $x^x$ bottoms at $\frac{1}{e}$ (value $e^{-1/e}$)
• Minimum distance point-to-curve: minimise $D^2$; the shortest join is normal to the curve.

Syllabus reminder: JEE Main tests rate of change, monotonicity, maxima-minima only; tangents/normals and Rolle's/Lagrange's MVT are JEE Advanced territory (see JEE Corner).

Last-Minute Mistake Checklist

Before the exam, scan this list — each item is a real mark lost by thousands of students every year:

1. Decreasing quantities need negative rates — "length decreasing at 3 cm/min" means $\frac{dx}{dt} = -3$.
2. Differentiate first, substitute after. Plugging $r = 10$ before differentiating freezes the variable and gives 0.
3. Monotonic intervals are reported separately, never as a union with one verdict.
4. Critical points include non-differentiable points — corners of $|x|$-type functions.
5. Second derivative test failing ($f'' = 0$) is not a conclusion — switch to the first derivative test.
6. Endpoints belong in every absolute-extrema table. The champion is an endpoint surprisingly often.
7. Reject impossible roots with a reason (negative lengths, cuts larger than the sheet) — there's a mark for it.
8. Answer the question asked — dimensions AND the max/min value, with units.
9. In MCQs, verify with a cheap check: AM-GM for minima, a nearby test value for monotonicity, symmetry for two-answer distance problems.
10. Budget time: a Board 5-marker deserves 8-9 minutes; a JEE question 2 minutes. If the setup isn't clear in 30 seconds, skip and return.

Done revising? Take the Section 8 mock drill under exam timing — that's the real test of readiness.