RD Sharma Class 12 Solutions Chapter 18 - Maxima and Minima (Ex 18.2) Exercise 18.2

Important Definitions Covered in This Chapter

1. 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 a 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. The number f(c), in this case, is called the minimum value of f in I and the point c, in this case, is called a point of minimum value of f in I.

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

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

Important Theorems 

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

(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 point of inflection.

(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)=0 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.

Theorem: Let f be a continuous function on an interval I = [a, b]. Then, f has the absolute maximum value and f attains it at least once in I. Also, f has the absolute minimum value and attains it at least once in I.

Theorem:  Let f be a differentiable function on a closed interval I and let c be any interior point of I. Then,

(i) f ′(c) = 0 if f attains its absolute maximum value at c.

(ii) f ′(c) = 0 if f attains its absolute minimum value at c.