Notes On Maxima and Minima - CBSE Class 12 Maths
Let x1,x2 ∈ D f(a) > f(x1) and f(a) > f(x2) A function is said to have a maximum value at a point 'a' in its domain D if f(a) ≥ f(x), ∀x ∈ D. f(a) is called the maximum value of f. "a" is called the point of maximum value of f. Let x1,x2 ∈ D f(a) < f(x1) and f(a) < f(x2) A function is said to have a minimum value at a point 'a' in its domain D if f(a) ≤ f(x), ∀x ∈ D. f(a) is called the minimum value of f. "a" is called the point of minimum value of f. Extreme value of function f: Function f(x) is said to have an extreme value in its domain D if there exists a point a ∈ D such that f (a) is either the maximum value or the minimum value of f in D. The number f (a) is called an extreme value of f in D, and point 'a' is called an extreme point. Consider f(x) = 2x2 ,∀x ∈ R         x   -2   -1   0   1   2     f(x) = 2x2   8   2   0   2   8 The ordered pairs are (-2,8),(-1,2),(0,0),(1,2) and (2,8). From the graph, we have f(x) = 0 if x = 0 and f(x) ≥ 0 ∀x ∈ R The minimum value of f is 0, and the point of minimum value of f is x = 0. From the graph of the function, f has no maximum value, and hence, no point of maximum value of f in R. Suppose the domain of f is restricted to [- 2, 1]. The maximum value of the function, f(-2) = 2(-2)2 = 8, which is at x = -2 Note: A function has an extreme value at a point even if it is not differentiable at that point. Result: Every increasing or decreasing function assumes its maximum or minimum value at the end points of the domain of definition of the function. Or Every continuous function on a closed interval has a maximum and a minimum value. Local Maxima and Local Minima The graph of the function has minimum values in some neighbourhood of points Q and S, and has maximum values in some neighbourhood of points P, R and T. The x-coordinates of points P, R and T are called the points of local maximum, and the y-coordinates of points P, R and T, are the local maximum values of f(x). Similarly, the x-coordinates of points Q and S are called the points of local minimum, and the y-coordinates of points Q and S, are called the local minimum values of f(x). The local maximum value of a function is referred to as the local maxima of that function. The local minimum value of a function is referred to as the local minima of that function. Let f be a real valued function defined on domain D. Let c ∈ D Then i) c is called a point of local maxima if there is an h > 0 such that f(c) ≥ f(x), ∀ x ∈ (c-h,c+h). The value f(c) is called the local maximum value of f. ii) c is called a point of local minima if there is a h > 0 such that f(c) ≤ f(x), ∀ x ∈ (c-h,c+h). The value f(c) is called the local minimum value of f. Let the point of local maximum value of f in the graph be x = a. The function f is increasing in the interval (a - h, a) and decreasing in the interval (a , a+h ), where h > 0. If f is increasing, then f '(x) > 0, and if f is decreasing, then f '(x) < 0. If f is neither decreasing nor increasing, then f '(x) = 0. i.e. f '(a) = 0. Let the point of local minimum value of f in the graph be x = a. Function f is decreasing in the interval (a - h, a) and increasing in the interval (a , a+h ), where h > 0. we have f '(a) = 0 Theorem: Let f be a real valued function defined on an open interval I. Suppose point a is any arbitrary point in I. If f has a local maxima or a local minima at x = a, then either f '(a) = 0, or f is not differentiable at a. However, the converse need not be true, i.e. a point at which the derivative vanishes need not be the point of local maxima or local minima. Every continuous function on a closed interval has a maximum and a minimum value.

#### Summary

Let x1,x2 ∈ D f(a) > f(x1) and f(a) > f(x2) A function is said to have a maximum value at a point 'a' in its domain D if f(a) ≥ f(x), ∀x ∈ D. f(a) is called the maximum value of f. "a" is called the point of maximum value of f. Let x1,x2 ∈ D f(a) < f(x1) and f(a) < f(x2) A function is said to have a minimum value at a point 'a' in its domain D if f(a) ≤ f(x), ∀x ∈ D. f(a) is called the minimum value of f. "a" is called the point of minimum value of f. Extreme value of function f: Function f(x) is said to have an extreme value in its domain D if there exists a point a ∈ D such that f (a) is either the maximum value or the minimum value of f in D. The number f (a) is called an extreme value of f in D, and point 'a' is called an extreme point. Consider f(x) = 2x2 ,∀x ∈ R         x   -2   -1   0   1   2     f(x) = 2x2   8   2   0   2   8 The ordered pairs are (-2,8),(-1,2),(0,0),(1,2) and (2,8). From the graph, we have f(x) = 0 if x = 0 and f(x) ≥ 0 ∀x ∈ R The minimum value of f is 0, and the point of minimum value of f is x = 0. From the graph of the function, f has no maximum value, and hence, no point of maximum value of f in R. Suppose the domain of f is restricted to [- 2, 1]. The maximum value of the function, f(-2) = 2(-2)2 = 8, which is at x = -2 Note: A function has an extreme value at a point even if it is not differentiable at that point. Result: Every increasing or decreasing function assumes its maximum or minimum value at the end points of the domain of definition of the function. Or Every continuous function on a closed interval has a maximum and a minimum value. Local Maxima and Local Minima The graph of the function has minimum values in some neighbourhood of points Q and S, and has maximum values in some neighbourhood of points P, R and T. The x-coordinates of points P, R and T are called the points of local maximum, and the y-coordinates of points P, R and T, are the local maximum values of f(x). Similarly, the x-coordinates of points Q and S are called the points of local minimum, and the y-coordinates of points Q and S, are called the local minimum values of f(x). The local maximum value of a function is referred to as the local maxima of that function. The local minimum value of a function is referred to as the local minima of that function. Let f be a real valued function defined on domain D. Let c ∈ D Then i) c is called a point of local maxima if there is an h > 0 such that f(c) ≥ f(x), ∀ x ∈ (c-h,c+h). The value f(c) is called the local maximum value of f. ii) c is called a point of local minima if there is a h > 0 such that f(c) ≤ f(x), ∀ x ∈ (c-h,c+h). The value f(c) is called the local minimum value of f. Let the point of local maximum value of f in the graph be x = a. The function f is increasing in the interval (a - h, a) and decreasing in the interval (a , a+h ), where h > 0. If f is increasing, then f '(x) > 0, and if f is decreasing, then f '(x) < 0. If f is neither decreasing nor increasing, then f '(x) = 0. i.e. f '(a) = 0. Let the point of local minimum value of f in the graph be x = a. Function f is decreasing in the interval (a - h, a) and increasing in the interval (a , a+h ), where h > 0. we have f '(a) = 0 Theorem: Let f be a real valued function defined on an open interval I. Suppose point a is any arbitrary point in I. If f has a local maxima or a local minima at x = a, then either f '(a) = 0, or f is not differentiable at a. However, the converse need not be true, i.e. a point at which the derivative vanishes need not be the point of local maxima or local minima. Every continuous function on a closed interval has a maximum and a minimum value.

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