Maxima and Minima

Let f be a function defined on an interval I. Then 

(a) f is said to have a maximum value in I if there exists a point c in I such that f(c)>f(x) for all x ∈ I.
The number f(c) is called the maximum value of f in I, and the point c is called the point of maximum value of f in I. 

(b) f is said to have a minimum value in I if there exists a point c in I such that f(c) < f(x) for all x ∈ I. 
In this case, the number f(c) is called the minimum value of f in I, and the point c, in this case, is called the point of minimum value of f in I. 

(c) f is said to have an extreme value in I if a point c exists in I such that f (c) is either a maximum value or a minimum value of f in I. 
In this case, the number f(c) is an extreme value of f in I, and the point c is an extreme point.

Remark: In the following Fig, the graphs of certain particular functions help us find the maximum value and minimum value at a point. In fact, through graphs, we can even find a function's maximum/minimum value at a point at which it is not even differentiable.

Let f be a real-valued function and c be an interior point in the domain of f. Then 

(a) c is called a point of local maxima if there is a h > 0 such that 
f(c) ≥ f(x), for all x in (c – h, c + h), x ≠ c 
The value f(c) is called the local maximum value of f. 

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

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

Geometrically, the above definition states that if x = c is a point of local maxima of f, then the graph of f around c will be as shown in Fig. Note that the function f is increasing (i.e., f′(x) > 0) in the interval (c – h, c) and decreasing (i.e., f′(x) < 0) in the interval (c, c + h).
This suggests that f′(c) must be zero.

Similarly, suppose c is a point of local minima of f. In that case, the graph of f around c will be as shown in the following Fig. Here, f is decreasing (i.e., f′(x) < 0) in the interval (c – h, c) and increasing (i.e., f′(x) > 0) in the interval (c, c + h). This again suggests that f′(c) must be zero. 
The above discussion leads us to the following theorem (without proof). 

Let f be a function defined on an open interval I. Suppose c ∈ I be any point. If f has a local maxima or a local minima at x = c, then either f′(c) = 0 or f is not differentiable at c. 



Remark: The converse of the above theorem need not be true; that is, a point at which the derivative vanishes need not be a point of local maxima or local minima. For example, if f(x) = `x^3`, then f′(x) = `3x^2` and so f′(0) = 0. But 0 is neither a point of local maxima nor a point of local minima in the above Fig. 

(First Derivative Test):
Let f be a function defined on an open interval I. Let f be continuous at a critical point c in I. Then 

(i) If f′(x) changes sign from positive to negative as x increases through c, i.e., if f′(x) > 0 at every point sufficiently close to and to the left of c, and f′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima. 

(ii) If f′(x) changes sign from negative to positive as x increases through c, i.e., if f′(x) < 0 at every point sufficiently close to and to the left of c, and f′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima. 

(iii) If f′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. In fact, such a point is called a point of inflexion, Fig.

Second Derivative Test:
Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then 

(i) x = c is a point of local maxima if f′(c) = 0 and f″(c) < 0 The value f (c) is the local maximum value of f.

(ii) x = c is a point of local minima if f'(c) =  and f″(c) > 0. In this case, f (c) is the local minimum value of f. 

(iii) The test fails if f′(c) = 0 and f″(c) = 0. In this case, we go back to the first derivative test and find whether c is a point of local maxima, local minima, or a point of inflexion.