Maxima and Minima: Overview, Questions, Preparation

Let f be a function and Let c be an interior point in its domain, then

(a) c is called a point of local maxima if there is an h > 0 such that f(c) ≥ f(x), for all

x $\in$ (c – h, c + h).

The number f(c) is called the local maximum value of f.

(b) c is called a point of local minima if there is an h > 0 such that f(c) ≤ f(x), for all

x $\in$ (c – h, c + h).

The number f(c) is called the local minimum value of f.

Maxima and Minima are generally calculated using derivative method, and here’s how:

Initially, calculate the critical points.

Next, apply the first derivative test:

• If f′(x) changes from positive to negative, that point is a local maximum.
• If f′(x) changes from negative to positive, that point is a local minimum.
• If f′(x) does not change any sign, the point is neither maxima nor minima.

Finally, calculate the second derivative test:

• If f′′(a)>0, it is a local minimum.
• If f′′(a)<0, it is a local maximum.
• If f′′(a)=0, the test is inconclusive.

Extrema is a general term that is used to refer to both maxima and minima. Finding the extrema of a function means finding both maxima and minima of the function. So don’t get confused in the paper if you are asked this term instead of maxima and minima. 

Types of Extrema

There are mainly two types of extrema:

1. Local (Relative) Maxima/Minima: These are referred to the highest or lowest points within a specific neighbourhood or interval on the function's graph. 
2. Absolute (Global) Maxima/Minima: These are referred to the highest or lowest points on the entire domain of the function’s graph. 

Let us understand the concept of maxima and minima through a practise question which will help clear our base.

Question: Find the maxima and minima of f(x) = x^2 – 4x + 3

Answer:

f′(x)=2x−4
→ Set f′(x)= 0 → x=2

f′′(x)=2>0 → So x=2 is a local minimum.

Value at x=2:

F(2) = (2)^2 – 4(2) + 3 = -1

Hence, at x = 2, minimum value is -1 (And no maxima)