# Test for the local Minimum/ Maximum – First Derivative Test

## Test for the local Minimum/ Maximum – First Derivative Test

Test for the local maximum/ minimum at x = a, if f(x) is differentiable at x = a:

If f(x) is differentiable at x = a   and if it is a critical point of the function (i.e., f’(a) = 0), then we have the following three tests to decided whether f(x) has a local maximum or local minimum or neither at x = a.

(i) First Derivative Test:

If f’(a) = 0 and f’(x) changes its sign while passing through the point x = a, then

(a) f(x) would have a local maximum at x = a, if f’(a – 0) > 0 and f’(a + 0) < 0. It means that f’(x) should changes its sign from positive to negative.

(b) f(x) would have local minimum at x = a, if f’(a – 0) < 0 and f’(a + 0) > 0. It means that f’(x) should changes its sign from negative to positive

(c)If f’(x) does not change its sign while passing through x = a, then f(x) would have neither a maximum nor minimum at x = a

Since at x = -1, we have local maximum

⇒ f(x) = -2

and at x = 1, we have local minimum

⇒ fmin(x) = 2

Example: Let f(x) = x + 1/x, x ≠ 0, then at which f(x) assume maximum and minimum are respectively.

Solution: Given that,

f(x) = x + 1/x, x≠0

differentiation with respect to x

f’(x) = 1 + (-1/x²)

= 1 – 1/x²

= (x² – 1)/x²

= ((x + 1) (x – 1))/x²

= at x = -1, we have local maximum

⇒ fmax (x) = -2

And at x = 1, we have local minimum

⇒ fmin(x) = 2.