Answer :

Solution :

Concept:

Consider a function y = f(x) on a defined interval of x. The function attains extreme values (the value can be maximum or minimum or both).

For maxima:

Local maxima: A point is the local maxima of a function if there is some other point where the maximum value is greater than the local maxima but that point doesn’t exist nearby local maxima.

Global maxima: It is the point where there is no other point has in the domain for which function has more value than global maxima.

For minima:

Local minima: A point is the local minima of a function if there is some other point where the minimum value is less than the local minima but that point doesn’t exist nearby local minima.

Global minima: It is the point where there is no other point has in the domain for which function has less value than global minima.

Stationary Points:

Points where the derivative of the function is zero i.e., f’(x) = 0.

The points can be: Inflection point ,Local maxima, Local minima

Second derivative test: Let the function has a stationary point x = a

if ${\left(\frac{d^{2}f}{d{x}^2}\right)}_{x=a}<0$ x=-1 and x=2 are extreme points

If ${\left(\frac{d^{2}f}{d{x}^2}\right)}_{x=a}>0$ then x = a, is a point of minima.

Application:

Given f(x) is a real -valued function such that f'(x₀) = 0 for some x₀ ∈ (0, 1) Also given f"(x) > 0 for all x ∈ (0, 1) So,

Then f(x) has exactly one local minimum in (0, 1), called the point of minima.