Let the tangent to the graph of y = f(x) at the point x = a be parallel to the x-axis, and let f'(a - h) > 0 and f'(a + h) < 0, where h is very small positive quantity. Then, the ordinate at the point x = a is

a maximum

We have,

f'(a - h) > 0

⇒ f(x) is increasing in the left neighbourhood of x = a

f'(a + h) < 0

⇒ f(x) is decreasing in the right neighbourhood of x = a

∴ f(a) is a local maximum value of f(x).