If y = f(x) is continuous on the interval [a, b] max y occurs at a point c, and f'(c) ≠ 0, f'(c) exists than:

c is equal to a or b

The correct answer is Option (3) → c is equal to a or b

According to calculus: If a function f(x) attains a local maximum or minimum at an interior point c∈(a,b) and if derivative of f(c) exists then that must be equal to zero.

However, this is not the case.

Thus, c must be an endpoint.

as $f'(c)≠0$ ⇒ max occurs at either a or b

$c=a$ or $c=b$