Maxima and Minima in Calculus - Practice Questions & MCQ

Maxima and Minima of a Function

Let y = f(x) be a real function defined at x = a. Then the function f(x) is said to have a maximum value at x = a, if f(x) ≤ f(a)  ∀ a ∈ R.

And also the function f(x) is said to have a minimum value at x = a, if f(x) ≥ f(a)  ∀ a ∈ R

Concept of Local Maxima and Local Minima 

The function f(x) is said to have a local maxima (or maxima) at a point ‘a’ if the value of f(x) at ‘a’  is greater than its values for all x in a small neighborhood of ‘a’ .

In other words, f(x) has a maxima at x = ‘a’, if f(a + h) ≤ f(a) and f(a - h) ≤ f(a), where h > 0 (very small quantity).

The function f(x) is said to have a local minima (or minima) at a point ‘b’ if the value of f(x) at ‘b’  is less than its values for all x in a small neighborhood of ‘b’ .

In other words, f(x) has a maximum at x = ‘b’, if f(b + h) ≥ f(b) and f(b - h) ≥ f(b), where h > 0 (very small quantity).