Summary of Class 12 Maths Applications of Derivatives
Rate of Change of Quantities
Let y = f(x) be a function of x,
where x is an independent variable. Assuming that y can be
differentiated with respect to x, the derivative 
at x = x0
or f’(x0) denotes the rate of change of y with respect to x.
If y = f(t) and x = f(t) be the two functions of t, then by Chain Rule, the rate of change of y with
respect to x is given by:
= (
)/(
) provided ≠ 0.
Thus, we can calculate the rate of change of y with respect to x, by
finding the rate of change of both y and x with respect to t, if is non-zero.
A point c in the interval (a, b) is
called a critical point of y = f(t), if 
= 0 at t
= c.
Increasing and Decreasing Functions
Increasing Functions
Let I be an interval contained in the
domain of a real valued function f.
Then f is said to be
(i) increasing on I, if x1
< x2 in I ⟹ f (x1) ≤ f (x2) for all x1,
x2 
I.
(ii) strictly increasing on I, if x1
< x2 in I ⟹ f (x1) < f (x2)
for all x1, x2 I.
Decreasing Functions
Let I be an interval contained in the
domain of a real valued function f.
Then f is said to be
(i) decreasing on I, if x1
< x2 in I ⟹ f (x1) ≥ f (x2) for
all x1, x2 I.
(ii) strictly decreasing on I, if x1
< x2 in I ⟹ f (x1) > f (x2)
for all x1, x2 I.
Theorem: Let f(x) be a
continuous function on [a, b] and differentiable on (a, b). Then:
(i) f is increasing in [a,
b], if f’(x) 
0 for each x 
(a, b).
(ii) f is decreasing in [a,
b], if f’(x) 
0 for each x (a, b).
(iii)
f is constant in [a, b], if f’(x) = 0 for each
x 
(a, b).
Maximum, Minimum and Extreme Values
Let
f be a function defined on an interval I. Then:
(a)
f is said to have a maximum value in I, if there exists a point c
in I such that f (c) > f (x), for all x ∈ I.
The
number f (c) is called the maximum value of f in I and the
point c is called a point of maximum value in I.
(b)
f is said to have a minimum value in I, if there exists a point c
in I such that f (c) < f (x), for all x ∈ I.
The
number f (c), in this case, is called the minimum value of f in
I and the point c is called a point of minimum value in I.
(c)
f is said to have an extreme value in I, if there exists a point c
in I such that f (c) is either a maximum value or a minimum
value of f in I.
The number f (c), in this case, is called an extreme value of f in I and the point c is called an extreme point.
Local Maxima and Local Minima
Let
f be a real valued function and let c be an interior point in the domain
of f. Then:
(a) c is called a point of local
maxima, if there is an h > 0
such that
f (c) ≥ f (x), for all
x in (c – h, c + h), x ≠ c
The
value f(c) is called the local maximum value of f.
(b) c is called a point of local
minima, if there is an h >
0 such that
f (c) ≤ f (x), for all
x in (c – h, c + h)
The value f(c) is called the local minimum value of f.
First Derivative Test
Let
f be a function defined on an open interval I
and let f be continuous at a critical point c in I. Then:
(i) if f ′(x)
changes sign from positive to negative as x increases through c, i.e., if f
′(x) > 0 at every point sufficiently close to and to the left of c, and f
′(x) < 0 at every point sufficiently close to and to the right of c, then c
is a point of local maxima.
(ii) if f ′(x)
changes sign from negative to positive as x increases through c, i.e., if f
′(x) < 0 at every point sufficiently close to and to the left of c, and f
′(x) > 0 at every point sufficiently close to and to the right of c, then c
is a point of local minima.
(iii) if f ′(x) does not change sign as x increases through c, then c is neither a point of local maximum nor a point of local minimum. In fact, such a point is called point of inflection.
Second Derivative Test
Let
f be a function defined on an interval I and c ∈ I. Let f
be twice differentiable at c. Then:
(i) x = c is a point of
local maxima, if f′(c) = 0 and f″(c) < 0. The value f (c)
is local maximum value of f.
(ii) x = c is a point of
local minima, if f’(c) = 0 and f″(c) > 0. In this case,
f(c) is local minimum value of f.
(iii) The test fails, if
f′(c) = 0 and f″(c) = 0.
In
this case, we go back to the first derivative test and find whether c is a
point of local maxima, local minima or a point of inflection.
Steps
Involved in Finding the Local Maxima and Local Minima
Below are
the steps involved in finding the local maxima and local minima of a given
function f(x) using the first derivative test.
Step 1: Evaluate the first derivative of f(x),
i.e., f’(x).
Step 2: Identify the critical points, i.e., value(s)
of c by assuming f’(x) = 0.
Step 3: Analyse the intervals where the given
function is increasing or decreasing.
Step 4: Determine the extreme points, i.e., local maxima or local minima.
Absolute Maxima and Absolute Minima
Let f be a continuous
function on an interval I = [a, b]. Then f has the
absolute maximum value and f attains it at least once in I. Similarly, f has the absolute minimum value and f attains it at least once in I.