Let f be a function defined on an interval I. Let a ∈ I be any element .Let f ''(a) exist.
(i) If f '(a) = 0 and f ''(a) < 0, then a is the point of local maxima and f (a) is the local maximum value of f.
(ii) If f '(a) = 0 and f ''(a) > 0, then a is the point of local minima and f (a) is the local minimum value of f.
iii) If f '(a) = 0 and f ''(a) = 0, then the test fails.
Given f '(a) = 0 f "(a) < 0 then
f "(a) = < 0
⇒ f "(a) = < 0
⇒ f "(a) = < 0
f '(a + h)/h < 0
Case I: Suppose h < 0
⇒ f '(a + h) > 0
Case II: Suppose h > 0
⇒ f '(a + h) < 0
Hence, f has local maximum at point a.
Local maximum value = f(a)
If f '(a ) = 0 and f "(a) = 0, then the second derivative test fails.
Working rule to find extreme values:
Step1: Find all the points where f'(x) = 0 ⇒ x = x1,x2,x3,.....
Step2: Find f "(x)
Step3: Find f "(x1),f "(x2) and f "(x3)....
Step4: If f "(x1) < 0, then the function has a local maximum value at x1.
Local maximum value = f(x1)
If f "(x1) > 0, then the function has a local minimum value at x1 .
Local minimum value = f(x1).
Let f be a function defined on an interval I. Let a ∈ I be any element .Let f ''(a) exist.
(i) If f '(a) = 0 and f ''(a) < 0, then a is the point of local maxima and f (a) is the local maximum value of f.
(ii) If f '(a) = 0 and f ''(a) > 0, then a is the point of local minima and f (a) is the local minimum value of f.
iii) If f '(a) = 0 and f ''(a) = 0, then the test fails.
Given f '(a) = 0 f "(a) < 0 then
f "(a) = < 0
⇒ f "(a) = < 0
⇒ f "(a) = < 0
f '(a + h)/h < 0
Case I: Suppose h < 0
⇒ f '(a + h) > 0
Case II: Suppose h > 0
⇒ f '(a + h) < 0
Hence, f has local maximum at point a.
Local maximum value = f(a)
If f '(a ) = 0 and f "(a) = 0, then the second derivative test fails.
Working rule to find extreme values:
Step1: Find all the points where f'(x) = 0 ⇒ x = x1,x2,x3,.....
Step2: Find f "(x)
Step3: Find f "(x1),f "(x2) and f "(x3)....
Step4: If f "(x1) < 0, then the function has a local maximum value at x1.
Local maximum value = f(x1)
If f "(x1) > 0, then the function has a local minimum value at x1 .
Local minimum value = f(x1).