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.