Maximum and Minimum value of a function in a closed interval

The graph of a continuous function defined on a closed interval [x₁ , x₈].

The function has a local minima at x = x₂, x₄ and x₆.

Local minimum values are f(x₂), f(x₄) and f(x₆).

The function has a local maxima at x = x₃, x₅ and x₇

Local maxima values are f(x₃), f(x₅) and f(x₇)

Absolute minimum value = Minimum {f(x₁),f(x₂),f(x₃),f(x₄),f(x₅),f(x₆),f(x₇),f(x₈)}

Absolute maximum value = Maximum {f(x₁),f(x₂),f(x₃),f(x₄),f(x₅),f(x₆),f(x₇),f(x₈)}

Absolute minimum value: A function f has absolute minimum value at point c ∈ I if f(c) ≤ f(x) ∀x ∈ I, where I is the domain of f.

The number f(c) is called the absolute minimum value of f on I.

Absolute maximum value: A function f has absolute maximum value at point c ∈ I if f(c) ≥ f(x) ∀x ∈ I, where I is the domain of f.

The number f(c) is called the absolute maximum value of f on I.

Important results

1. Let f be a continuous function on [a, b]. Then f attains the absolute maximum value and absolute minimum value at least once in the closed interval [a, b].

2. Let f be a differentiable function on [a, b]. Then f is said to attain absolute maximum value (or absolute minimum value) at a point α ∈ [a,b], if f '(α) = 0.

3. Local maxima and local minima occur alternately, i.e. between every two local maxima, there is one local minima, and vice versa.

4. Local minima at some points may be greater than local maxima at some other points.

Working rule:

Consider a function f defined on [a,b].

Step 1: Find the derivative of the function,f.

Step 2: Put f '(x) = 0 to find the all the critical points of the function.

Step 3: Find the value of the function at all the critical points and also find f(a), f(b) .

Step 4:

Absolute minimum value = Minimum{Values of the function at all critical points, f(a), f(b)}

Absolute maximum value = Maximum{Values of the function at all critical points, f(a), f(b)}