Let x_{1},x_{2 }∈ D f(a) > f(x_{1}) and f(a) > f(x_{2})
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 x_{1},x_{2 }∈ D f(a) < f(x_{1}) and f(a) < f(x_{2})
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) = 2x^{2 },∀x ∈ R
x |
-2 |
-1 |
0 |
1 |
2 |
f(x) = 2x^{2} |
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.
Let x_{1},x_{2 }∈ D f(a) > f(x_{1}) and f(a) > f(x_{2})
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 x_{1},x_{2 }∈ D f(a) < f(x_{1}) and f(a) < f(x_{2})
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) = 2x^{2 },∀x ∈ R
x |
-2 |
-1 |
0 |
1 |
2 |
f(x) = 2x^{2} |
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.