Maxima and Minima
Notes Transcription
Definitions
Maximum
A function f (x) is said to be maximum at x = a if there exists a very small positive
number δ such that for all values of h in the interval (−δ, δ) (where h ̸= 0):
f (a + h) < f (a)
Minimum
A function f (x) is said to be minimum at x = a if there exists a very small positive
number δ such that for all values of h in the interval (−δ, δ) (where h ̸= 0):
f (a + h) > f (a)
Conditions for Maxima and Minima
Necessary Condition
The necessary condition for f (x) to be a maximum or a minimum at x = a is that:
f ′ (a) = 0
Analysis using Taylor’s Theorem
Let f (x) be a given function that can be expanded by Taylor’s theorem in the neighbor-
hood of x = a. Then:
h2 ′′ hn
f (a + h) = f (a) + hf ′ (a) + f (a) + · · · + f n (a)
2! n!
h2 ′′
⇒ f (a + h) − f (a) = hf ′ (a) + f (a) + . . .
2!
For f (x) to be a maximum or minimum at x = a, the sign of f (a + h) − f (a) must
be invariant for small values of h. When h is sufficiently small, the sign is governed by
the term of the lowest degree in h. If f ′ (a) ̸= 0, the sign of hf ′ (a) changes with the sign
of h. Therefore, for a maximum or minimum, we must have f ′ (a) = 0.
1
, Sufficient Condition
If f ′ (a) = 0, the expansion becomes:
h2 ′′ h3
f (a + h) − f (a) = f (a) + f ′′′ (a) + . . .
2! 3!
Since h2 is always positive:
• For **Maxima**: f (a + h) − f (a) < 0 =⇒ f ′′ (a) < 0 (Negative value).
• For **Minima**: f (a + h) − f (a) > 0 =⇒ f ′′ (a) > 0 (Positive value).
Working Rule for Maxima and Minima
1. Find f ′ (x) and equate it to zero (f ′ (x) = 0).
2. Solve the resulting equation for x. Let the roots be a1 , a2 , . . . .
3. Find f ′′ (x) and substitute x = a1 , a2 , . . . into it.
4. Check the sign of f ′′ (a):
• If f ′′ (a) > 0, then it is a **Minima**.
• If f ′′ (a) < 0, then it is a **Maxima**.
5. If f ′′ (a) = 0, then find f ′′′ (x). If f ′′′ (a) ̸= 0, the function has neither maxima nor
minima (inflection point). If f ′′′ (a) = 0, continue to the next derivative.
2
Notes Transcription
Definitions
Maximum
A function f (x) is said to be maximum at x = a if there exists a very small positive
number δ such that for all values of h in the interval (−δ, δ) (where h ̸= 0):
f (a + h) < f (a)
Minimum
A function f (x) is said to be minimum at x = a if there exists a very small positive
number δ such that for all values of h in the interval (−δ, δ) (where h ̸= 0):
f (a + h) > f (a)
Conditions for Maxima and Minima
Necessary Condition
The necessary condition for f (x) to be a maximum or a minimum at x = a is that:
f ′ (a) = 0
Analysis using Taylor’s Theorem
Let f (x) be a given function that can be expanded by Taylor’s theorem in the neighbor-
hood of x = a. Then:
h2 ′′ hn
f (a + h) = f (a) + hf ′ (a) + f (a) + · · · + f n (a)
2! n!
h2 ′′
⇒ f (a + h) − f (a) = hf ′ (a) + f (a) + . . .
2!
For f (x) to be a maximum or minimum at x = a, the sign of f (a + h) − f (a) must
be invariant for small values of h. When h is sufficiently small, the sign is governed by
the term of the lowest degree in h. If f ′ (a) ̸= 0, the sign of hf ′ (a) changes with the sign
of h. Therefore, for a maximum or minimum, we must have f ′ (a) = 0.
1
, Sufficient Condition
If f ′ (a) = 0, the expansion becomes:
h2 ′′ h3
f (a + h) − f (a) = f (a) + f ′′′ (a) + . . .
2! 3!
Since h2 is always positive:
• For **Maxima**: f (a + h) − f (a) < 0 =⇒ f ′′ (a) < 0 (Negative value).
• For **Minima**: f (a + h) − f (a) > 0 =⇒ f ′′ (a) > 0 (Positive value).
Working Rule for Maxima and Minima
1. Find f ′ (x) and equate it to zero (f ′ (x) = 0).
2. Solve the resulting equation for x. Let the roots be a1 , a2 , . . . .
3. Find f ′′ (x) and substitute x = a1 , a2 , . . . into it.
4. Check the sign of f ′′ (a):
• If f ′′ (a) > 0, then it is a **Minima**.
• If f ′′ (a) < 0, then it is a **Maxima**.
5. If f ′′ (a) = 0, then find f ′′′ (x). If f ′′′ (a) ̸= 0, the function has neither maxima nor
minima (inflection point). If f ′′′ (a) = 0, continue to the next derivative.
2
