Monotonicity formulas

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Formulas: 26
Subject: Mathematics
Exam: JEE Advanced
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Monotonicity, every formula

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

Increasing function
$x_{1} < x_{2} \Rightarrow f (x_{1}) \leq f (x_{2})$
Strictly increasing
$x_{1} < x_{2} \Rightarrow f (x_{1}) < f (x_{2})$
Decreasing function
$x_{1} < x_{2} \Rightarrow f (x_{1}) \geq f (x_{2})$
Strictly decreasing
$x_{1} < x_{2} \Rightarrow f (x_{1}) > f (x_{2})$
Test: increasing
$f^{'} (x) \geq 0 \forall x \in (a, b)$
Test: strictly increasing
$f^{'} (x) > 0 \forall x \in (a, b)$
Test: decreasing
$f^{'} (x) \leq 0 \forall x \in (a, b)$
Test: strictly decreasing
$f^{'} (x) < 0 \forall x \in (a, b)$
Increasing at a point
$f (a - h) < f (a) < f (a + h)$
small h>0
Decreasing at a point
$f (a - h) > f (a) > f (a + h)$
Monotonic function
$either increasing or decreasing throughout its domain$
Strict monotonic ⇒ one-one
$strictly monotonic \Rightarrow injective (one-one)$
Inverse of increasing
$f strictly increasing \Rightarrow f^{- 1} strictly increasing$
Composite of monotone
$both same type \Rightarrow f \circ g increasing; opposite \Rightarrow decreasing$
Critical point
$f^{'} (a) = 0 or f^{'} (a) does not exist$
Stationary point
$f^{'} (a) = 0$
Rolle's theorem
$f (a) = f (b), f cts on [a, b], diff on (a, b) \Rightarrow f^{'} (c) = 0$
LMVT
$f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}$
LMVT alternative form
$f (a + h) = f (a) + h f^{'} (a + θ h), 0 < θ < 1$
Rolle as special LMVT
$f (a) = f (b) \Rightarrow f^{'} (c) = 0$
Cauchy's MVT
$\frac{f ( b ) - f ( a )}{ϕ ( b ) - ϕ ( a )} = \frac{f ^{'} ( c )}{ϕ ^{'} ( c )}$
Absolute maximum
$f (x) \leq f (a) \forall x \in D$
Absolute minimum
$f (x) \geq f (a) \forall x \in D$
Average rate = instant rate
$\frac{f ( b ) - f ( a )}{b - a} = f^{'} (c)$
geometric LMVT
Neither increasing nor decreasing
$f (a - h) > f (a) < f (a + h) or f (a - h) < f (a) > f (a + h)$
Monotone on interval from sign of f'
$find intervals where f^{'} (x) > 0 or < 0$

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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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