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Monotonicity formulas

Master Monotonicity through 26 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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Monotonicity, every formula

26 formulas, typeset and free. Print it, or keep it open beside your practice.

Increasing function
$x_{1} < x_{2} \Rightarrow f (x_{1}) \leq f (x_{2})$
Q1MCQIncreasing
$f$ is increasing on $(a,b)$ if $x_1<x_2$ implies:
- A
  $f(x_1)\le f(x_2)$
- B
  $f(x_1)\ge f(x_2)$
- C
  $f(x_1)=f(x_2)$
- D
  $f(x_1)>f(x_2)$
Strictly increasing
$x_{1} < x_{2} \Rightarrow f (x_{1}) < f (x_{2})$
Q1MCQStrictly increasing
$f$ is strictly increasing if $x_1<x_2$ implies:
- A
  $f(x_1)<f(x_2)$
- B
  $f(x_1)\le f(x_2)$
- C
  $f(x_1)>f(x_2)$
- D
  $f(x_1)=f(x_2)$
Decreasing function
$x_{1} < x_{2} \Rightarrow f (x_{1}) \geq f (x_{2})$
Q1MCQDecreasing
$f$ is decreasing if $x_1<x_2$ implies:
- A
  $f(x_1)\ge f(x_2)$
- B
  $f(x_1)\le f(x_2)$
- C
  $f(x_1)<f(x_2)$
- D
  $f(x_1)=f(x_2)$
Strictly decreasing
$x_{1} < x_{2} \Rightarrow f (x_{1}) > f (x_{2})$
Q1MCQStrictly decreasing
$f$ is strictly decreasing if $x_1<x_2$ implies:
- A
  $f(x_1)>f(x_2)$
- B
  $f(x_1)<f(x_2)$
- C
  $f(x_1)\ge f(x_2)$
- D
  $f(x_1)=f(x_2)$
Test: increasing
$f^{'} (x) \geq 0 \forall x \in (a, b)$
Q1MCQTest increasing
$f$ is increasing on $(a,b)$ if:
- A
  $f'(x)\ge0$
- B
  $f'(x)<0$
- C
  $f'(x)=0$
- D
  $f''(x)>0$
Test: strictly increasing
$f^{'} (x) > 0 \forall x \in (a, b)$
Q1MCQTest strictly increasing
$f(x)=x^{3}$ is strictly increasing because:
- A
  $f'(x)=3x^{2}>0$ for $x\neq0$
- B
  $f'(x)<0$
- C
  $f'(x)=0$ always
- D
  $f''(x)<0$
Test: decreasing
$f^{'} (x) \leq 0 \forall x \in (a, b)$
Q1MCQTest decreasing
$f$ is decreasing on $(a,b)$ if:
- A
  $f'(x)\le0$
- B
  $f'(x)\ge0$
- C
  $f'(x)>0$
- D
  $f''(x)>0$
Test: strictly decreasing
$f^{'} (x) < 0 \forall x \in (a, b)$
Q1MCQTest strictly decreasing
$f(x)=-x$ is strictly decreasing because:
- A
  $f'(x)=-1<0$
- B
  $f'(x)=1>0$
- C
  $f'(x)=0$
- D
  $f''(x)=0$
Increasing at a point
$f (a - h) < f (a) < f (a + h)$
small h>0
Q1MCQIncreasing at a point
$f$ is increasing at $a$ if for small $h>0$:
- A
  $f(a-h)<f(a)<f(a+h)$
- B
  $f(a-h)>f(a)>f(a+h)$
- C
  $f(a-h)=f(a)$
- D
  $f(a)=0$
Decreasing at a point
$f (a - h) > f (a) > f (a + h)$
Q1MCQDecreasing at a point
$f$ is decreasing at $a$ if for small $h>0$:
- A
  $f(a-h)>f(a)>f(a+h)$
- B
  $f(a-h)<f(a)<f(a+h)$
- C
  $f(a)=0$
- D
  $f'(a)=0$
Monotonic function
$either increasing or decreasing throughout its domain$
Q1MCQMonotonic
A monotonic function is one that is:
- A
  either increasing or decreasing throughout
- B
  constant
- C
  periodic
- D
  discontinuous
Strict monotonic ⇒ one-one
$strictly monotonic \Rightarrow injective (one-one)$
Q1MCQStrict ⇒ one-one
A strictly monotonic function is always:
- A
  one-one
- B
  many-one
- C
  even
- D
  periodic
Inverse of increasing
$f strictly increasing \Rightarrow f^{- 1} strictly increasing$
Q1MCQInverse increasing
If $f$ is strictly increasing, then $f^{-1}$ is:
- A
  strictly increasing
- B
  strictly decreasing
- C
  constant
- D
  undefined
Composite of monotone
$both same type \Rightarrow f \circ g increasing; opposite \Rightarrow decreasing$
Q1MCQComposite
If $f$ increasing and $g$ decreasing, then $f\circ g$ is:
- A
  decreasing
- B
  increasing
- C
  constant
- D
  non-monotonic
Critical point
$f^{'} (a) = 0 or f^{'} (a) does not exist$
Q1MCQCritical point
$x=a$ is a critical point if:
- A
  $f'(a)=0$ or $f'(a)$ does not exist
- B
  $f(a)=0$
- C
  $f''(a)=0$
- D
  $f(a)=1$
Stationary point
$f^{'} (a) = 0$
Q1MCQStationary point
$x=a$ is a stationary point if:
- A
  $f'(a)=0$
- B
  $f(a)=0$
- C
  $f''(a)=0$
- D
  $f'(a)$ undefined
Rolle's theorem
$f (a) = f (b), f cts on [a, b], diff on (a, b) \Rightarrow f^{'} (c) = 0$
Q1MCQRolle's theorem
Rolle's theorem requires, among the conditions:
- A
  $f(a)=f(b)$
- B
  $f(a)\neq f(b)$
- C
  $f'(a)=0$
- D
  $f$ constant
LMVT
$f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}$
Q1MCQLMVT
Lagrange's MVT gives some $c$ with $f'(c)$ equal to:
- A
  $\dfrac{f(b)-f(a)}{b-a}$
- B
  $f(b)-f(a)$
- C
  $0$
- D
  $\dfrac{f(a)+f(b)}{2}$
LMVT alternative form
$f (a + h) = f (a) + h f^{'} (a + θ h), 0 < θ < 1$
Q1MCQLMVT alt form
The alternative form of LMVT is:
- A
  $f(a+h)=f(a)+h\,f'(a+\theta h),\ 0<\theta<1$
- B
  $f(a+h)=f(a)$
- C
  $f'(c)=0$
- D
  $f(a+h)=hf'(a)$
Rolle as special LMVT
$f (a) = f (b) \Rightarrow f^{'} (c) = 0$
Q1MCQRolle from LMVT
Rolle's theorem is LMVT with the extra condition:
- A
  $f(a)=f(b)$
- B
  $f'(a)=0$
- C
  $b=a$
- D
  $f$ linear
Cauchy's MVT
$\frac{f ( b ) - f ( a )}{ϕ ( b ) - ϕ ( a )} = \frac{f ^{'} ( c )}{ϕ ^{'} ( c )}$
Q1MCQCauchy's MVT
Cauchy's MVT states:
- A
  $\dfrac{f(b)-f(a)}{\phi(b)-\phi(a)}=\dfrac{f'(c)}{\phi'(c)}$
- B
  $f'(c)=\phi'(c)$
- C
  $f(b)=\phi(b)$
- D
  $f'(c)=0$
Absolute maximum
$f (x) \leq f (a) \forall x \in D$
Q1MCQAbsolute maximum
$f$ 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 minimum
$f$ 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
  $f(a)=0$
Average rate = instant rate
$\frac{f ( b ) - f ( a )}{b - a} = f^{'} (c)$
geometric LMVT
Q1MCQGeometric LMVT
LMVT says the average rate of change equals the instantaneous rate at:
- A
  some interior point $c$
- B
  the endpoint $a$
- C
  the endpoint $b$
- D
  no point
Neither increasing nor decreasing
$f (a - h) > f (a) < f (a + h) or f (a - h) < f (a) > f (a + h)$
Q1MCQNeither incr nor decr
At $x=a$, if $f(a-h)>f(a)<f(a+h)$, then $f$ is:
- A
  neither increasing nor decreasing (a local min)
- B
  increasing
- C
  decreasing
- D
  constant
Monotone on interval from sign of f'
$find intervals where f^{'} (x) > 0 or < 0$
Q1MCQInterval of increase
For $f(x)=x^{2}-4x$, $f$ is increasing on:
- A
  $(2,\infty)$
- B
  $(-\infty,2)$
- C
  $(-\infty,\infty)$
- D
  $(0,4)$

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Monotonicity formulas

Monotonicity, every formula

Increasing function

Strictly increasing

Decreasing function

Strictly decreasing

Test: increasing

Test: strictly increasing

Test: decreasing

Test: strictly decreasing

Increasing at a point

Decreasing at a point

Monotonic function

Strict monotonic ⇒ one-one

Inverse of increasing

Composite of monotone

Critical point

Stationary point

Rolle's theorem

LMVT

LMVT alternative form

Rolle as special LMVT

Cauchy's MVT

Absolute maximum

Absolute minimum

Average rate = instant rate

Neither increasing nor decreasing

Monotone on interval from sign of f'

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