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# Second Derivative Test

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#### Second Derivative Test - Lesson Summary

Let f be a function defined on an interval I. Let a âˆˆ 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) =   lim h â†’ 0 f'(a+h) - f'(a) h   < 0

â‡’ f "(a) =   lim h â†’ 0 f'(a+h) - 0 h   < 0

â‡’ f "(a) =   lim h â†’ 0 f'(a+h) h   < 0

f '(a + h)/h < 0

Case I: Suppose h < 0

â‡’ f '(a + h) > 0

Case II: Suppose h > 0

â‡’ 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 â‡’ 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).