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Maxima AND Minima formulas

Master Maxima AND Minima through 32 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Maxima AND Minima, every formula

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Absolute maximum
$f (x) \leq f (a) \forall x \in D$
Q1MCQAbsolute max
$f$ has an absolute maximum at $a$ if:
- A
  $f(x)\le f(a)$ for all $x\in D$
- B
  $f(x)\ge f(a)$
- C
  $f'(a)=0$
- D
  $f(a)=0$
Absolute minimum
$f (x) \geq f (a) \forall x \in D$
Q1MCQAbsolute min
$f$ has an absolute minimum at $a$ if:
- A
  $f(x)\ge f(a)$ for all $x\in D$
- B
  $f(x)\le f(a)$
- C
  $f'(a)=0$
- D
  $f(a)=0$
Local maximum
$f (x) \leq f (a) \forall x \in (a - h, a + h)$
Q1MCQLocal max
$f$ has a local maximum at $a$ if, near $a$:
- A
  $f(x)\le f(a)$ on $(a-h,a+h)$
- B
  $f(x)\ge f(a)$
- C
  $f''(a)>0$
- D
  $f(a)=0$
Local minimum
$f (x) \geq f (a) \forall x \in (a - h, a + h)$
Q1MCQLocal min
$f$ has a local minimum at $a$ if, near $a$:
- A
  $f(x)\ge f(a)$ on $(a-h,a+h)$
- B
  $f(x)\le f(a)$
- C
  $f''(a)<0$
- D
  $f(a)=0$
Necessary condition
$f^{'} (a) = 0 (critical point)$
Q1MCQNecessary condition
A necessary condition for a local extremum of a differentiable $f$ at $a$ is:
- A
  $f'(a)=0$
- B
  $f(a)=0$
- C
  $f''(a)=0$
- D
  $f'(a)>0$
First derivative test (max)
$f^{'} changes + \to - at a$
Q1MCQ1st deriv test max
By the first derivative test, $f$ has a local maximum at $a$ if $f'$:
- A
  changes from $+$ to $-$
- B
  changes from $-$ to $+$
- C
  stays positive
- D
  stays negative
First derivative test (min)
$f^{'} changes - \to + at a$
Q1MCQ1st deriv test min
$f$ has a local minimum at $a$ if $f'$:
- A
  changes from $-$ to $+$
- B
  changes from $+$ to $-$
- C
  stays positive
- D
  stays negative
Second derivative test (max)
$f^{'} (a) = 0, f^{''} (a) < 0$
Q1MCQ2nd deriv test max
If $f'(a)=0$ and $f''(a)<0$, then $a$ is a:
- A
  local maximum
- B
  local minimum
- C
  point of inflection
- D
  saddle point
Second derivative test (min)
$f^{'} (a) = 0, f^{''} (a) > 0$
Q1MCQ2nd deriv test min
If $f'(a)=0$ and $f''(a)>0$, then $a$ is a:
- A
  local minimum
- B
  local maximum
- C
  point of inflection
- D
  saddle point
nth derivative test
$f^{'} (a) = \dots = f^{(n - 1)} (a) = 0, f^{(n)} (a) \neq = 0$
n even → extremum
Q1MCQnth derivative test
If $f'(a)=\cdots=f^{(n-1)}(a)=0,\ f^{(n)}(a)\neq0$, an extremum occurs when $n$ is:
- A
  even
- B
  odd
- C
  prime
- D
  any
Global extremum on [a,b]
$compare f (critical pts) and f (a), f (b)$
Point of inflection
$f^{''} (x_{0}) = 0 and f^{''} changes sign$
Q1MCQInflection
A point of inflection occurs where $f''(x_0)=0$ and $f''$:
- A
  changes sign
- B
  stays positive
- C
  stays negative
- D
  is undefined only
Convex (concave up)
$f^{''} (x) > 0$
Q1MCQConvex
$f$ is convex (concave up) where:
- A
  $f''(x)>0$
- B
  $f''(x)<0$
- C
  $f'(x)=0$
- D
  $f(x)=0$
Concave (concave down)
$f^{''} (x) < 0$
Q1MCQConcave
$f$ is concave (concave down) where:
- A
  $f''(x)<0$
- B
  $f''(x)>0$
- C
  $f'(x)=0$
- D
  $f(x)=0$
Min of (x-a)(x-b)
$- \frac{( a - b ) ^{2}}{4}$
Max of a cos²x + b sin²x
$max = a, min = b (a > b)$
Q1MCQMax a cos²+b sin²
The maximum value of $5\cos^{2}x+3\sin^{2}x$ is:
- A
  $5$
- B
  $3$
- C
  $8$
- D
  $4$
Min of a²sec²x + b²cosec²x
$(a + b)^{2}$
Q1MCQMin a²sec²+b²cosec²
The minimum value of $a^{2}\sec^{2}x+b^{2}\csc^{2}x$ is:
- A
  $(a+b)^{2}$
- B
  $2ab$
- C
  $a^{2}+b^{2}$
- D
  $(a-b)^{2}$
Min of a²sin²x + b²cosec²x
$2 ab$
Q1MCQMin a²sin²+b²cosec²
The minimum value of $a^{2}\sin^{2}x+b^{2}\csc^{2}x$ is:
- A
  $2ab$
- B
  $(a+b)^{2}$
- C
  $a^{2}+b^{2}$
- D
  $ab$
Min of a cot x + b tan x
$2 ab$
Q1MCQMin a cot+b tan
The minimum value of $a\cot x+b\tan x$ (for $a,b>0$, acute $x$) is:
- A
  $2\sqrt{ab}$
- B
  $a+b$
- C
  $2ab$
- D
  $\sqrt{a+b}$
Max of sinᵖθ cosᑫθ
$at θ = tan^{- 1} \frac{p}{q}$
Q1MCQMax sinᵖ cosᑫ
$\sin^{p}\theta\cos^{q}\theta$ attains its maximum at:
- A
  $\theta=\tan^{-1}\sqrt{\dfrac{p}{q}}$
- B
  $\theta=\dfrac{\pi}{4}$
- C
  $\theta=\tan^{-1}\sqrt{\dfrac{q}{p}}$
- D
  $\theta=\dfrac{\pi}{2}$
Product max (fixed sum)
$sum = k \Rightarrow product max at \frac{k}{2}, \frac{k}{2}$
Sum of squares min (fixed sum)
$min at \frac{k}{2}, \frac{k}{2}$
Max rectangle in circle
$square of side 2 r$
Q1MCQMax rectangle in circle
The maximum rectangle inscribed in a circle of radius $r$ is:
- A
  a square of side $\sqrt2\,r$
- B
  a rectangle of sides $r,2r$
- C
  a square of side $r$
- D
  a square of side $2r$
Max triangle in circle
$equilateral of side 3 r$
Q1MCQMax triangle in circle
The maximum triangle inscribed in a circle of radius $r$ is:
- A
  equilateral of side $\sqrt3\,r$
- B
  right-angled
- C
  isosceles of side $r$
- D
  equilateral of side $r$
Max sector area (fixed perimeter K)
$\frac{K ^{2}}{16}$
Q1MCQMax sector area
For a sector of fixed perimeter $K$, the maximum area is:
- A
  $\dfrac{K^{2}}{16}$
- B
  $\dfrac{K^{2}}{8}$
- C
  $\dfrac{K^{2}}{4}$
- D
  $\dfrac{K}{2}$
Least tangent-intercept (ellipse)
$a + b$
portion between axes
Q1MCQLeast tangent intercept
The least length of the tangent to $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ intercepted between the axes is:
- A
  $a+b$
- B
  $a-b$
- C
  $ab$
- D
  $\sqrt{a^{2}+b^{2}}$
Max rectangle in ellipse
$2 ab$
Q1MCQMax rectangle in ellipse
The maximum rectangle inscribed in $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ has area:
- A
  $2ab$
- B
  $ab$
- C
  $4ab$
- D
  $\dfrac{3\sqrt3}{4}ab$
Max isosceles triangle in ellipse
$\frac{3 3}{4} ab$
Q1MCQMax isosceles triangle in ellipse
The greatest isosceles triangle inscribed in $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ (vertex at a major-axis end) has area:
- A
  $\dfrac{3\sqrt3}{4}ab$
- B
  $2ab$
- C
  $\dfrac{\sqrt3}{4}ab$
- D
  $ab$
Open box from square sheet
$side removed = \frac{a}{6}$
Q1MCQOpen box from square
From a square sheet of side $a$, the side of the square removed for maximum box volume is:
- A
  $\dfrac{a}{6}$
- B
  $\dfrac{a}{4}$
- C
  $\dfrac{a}{2}$
- D
  $\dfrac{a}{3}$
Cone circumscribing sphere (max V)
$h = \frac{4 R}{3}, sin α = \frac{1}{3}$
Q1MCQCone in sphere
For a cone of maximum volume inscribed in a sphere of radius $R$, the height is:
- A
  $\dfrac{4R}{3}$
- B
  $\dfrac{2R}{3}$
- C
  $R$
- D
  $\dfrac{3R}{4}$
Least triangle by line through (p,q)
$2 pq$
with the axes
Q1MCQLeast triangle through (p,q)
The least area of the triangle formed by a line through $(p,q)$ and the axes is:
- A
  $2pq$
- B
  $pq$
- C
  $\dfrac{pq}{2}$
- D
  $4pq$
Sum of side & hypotenuse (max area)
$angle = 6 0^{\circ}$
Q1MCQSide + hypotenuse
If the sum of a side and the hypotenuse of a right triangle is fixed, the area is maximum when the angle between them is:
- A
  $60^{\circ}$
- B
  $45^{\circ}$
- C
  $30^{\circ}$
- D
  $90^{\circ}$

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Maxima AND Minima formulas

Maxima AND Minima, every formula

Absolute maximum

Absolute minimum

Local maximum

Local minimum

Necessary condition

First derivative test (max)

First derivative test (min)

Second derivative test (max)

Second derivative test (min)

nth derivative test

Global extremum on [a,b]

Point of inflection

Convex (concave up)

Concave (concave down)

Min of (x-a)(x-b)

Max of a cos²x + b sin²x

Min of a²sec²x + b²cosec²x

Min of a²sin²x + b²cosec²x

Min of a cot x + b tan x

Max of sinᵖθ cosᑫθ

Product max (fixed sum)

Sum of squares min (fixed sum)

Max rectangle in circle

Max triangle in circle

Max sector area (fixed perimeter K)

Least tangent-intercept (ellipse)

Max rectangle in ellipse

Max isosceles triangle in ellipse

Open box from square sheet

Cone circumscribing sphere (max V)

Least triangle by line through (p,q)

Sum of side & hypotenuse (max area)

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