**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

⇒
f_{min}(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

⇒ f_{max}
(x) = -2

And at x = 1, we have local minimum

⇒
f_{min}(x) = 2.