Let f be a real valued function such that f: I → R.
Let x1 ∈ I and h > 0
Consider (x1 - h, x1 + h)
(1)f is said to be increasing at x1, if f is increasing in (x1 - h, x1 + h)
(2)f is said to be strictly increasing at x1, if f is strictly increasing in (x1 - h, x1 + h)
(3)f is said to be decreasing at x1, if f is decreasing in (x1 - h, x1 + h)
(4)f is said to be strictly decreasing at x1, if f is strictly decreasing in (x1 - h, x1 + h)
Theorem: Let f be continuous on [a, b] and differentiable on (a, b) Then:
(i) f is increasing in [a, b], if f ' (x) ≥ 0, ∀ x ∈ (a, b)
(ii) f is decreasing in [a, b], if f ' (x) ≤ 0, ∀ x ∈ (a, b)
(iii) f is a constant function in [a, b] if f ' (x) = 0, ∀ x ∈ (a, b)
Proof:
Let x1, x2 ∈ [a, b] such that x1 < x2
By the mean value theorem, there exists c ∈ (x1, x2) such that f(c) = ....(1)
(i)Let f(c) ≥ 0
⇒ ≥ 0
⇒ ≥ 0 (Since x1 < x2)
⇒
Or
⇒ f is an increasing function.
(ii) Let f(c) ≤ 0
⇒ ≤ 0
(Since x1 < x2)
⇒
Or
⇒ f is a decreasing function.
(iii)Let f(c) = 0
⇒ = 0
⇒ = 0
⇒ , ∀ x1, x2 ∈ [a, b]
⇒
Some results:
(I)f is strictly increasing in (a, b) if f ' (x) > 0, ∀ x ∈ [a, b]
(II)f is strictly decreasing in (a, b) if f ' (x) < 0, ∀ x ∈ [a, b]
(III)f is increasing or decreasing on R if it is increasing or decreasing in every interval of R
Function f is increasing in [a, b], if f ' (x) ≥ 0, ∀ x ∈ (a, b).
Function f is decreasing in [a, b], if f ' (x) ≤ 0, ∀ x ∈ (a, b).
Function f is a constant function in [a, b] if f ' (x) = 0, ∀ x ∈ (a, b).
Example :
Show that the function f(x) = x3 – 6x2 + 15x + 3 is strictly increasing on the set of real numbers.
Sol :
Given, f(x) = x3 – 6x2 + 15x + 3
Differentiating with respect to x, we get
f(x) = 3x2 - 12x + 15
= 3(x2 - 4x + 5)
= 3(x2 - 4x + 4 + 1)
= 3((x - 2)2 + 1)
Since (x - 2)2 ≥ 1, ∀ x ∈ R
⇒ 3((x - 2)2 + 1) > 0, ∀ x ∈ R
⇒ f ' (x) > 0, ∀ x ∈ R
⇒ f(x) = x3 - 6x2 + 15x + 3 is strictly increasing on the set of real numbers.
Example :
Find the intervals in which the function, f, given by f(x) = x2 - 8x + 5 is:
(a) Strictly increasing
(b) Strictly decreasing
Sol :
Given f(x) = x2
Differentiating with respect to x, we get f(x) = 2x - 8
a) f is strictly increasing if f(x)
⇒ 2x - 8 > 0
⇒ 2x > 8
⇒ x > 4
⇒ x ∈ (4, ∞)
⇒ f is strictly increasing in (4, ∞)
b) f is strictly decreasing if f(x) < 0
⇒ 2x - 8 < 0
⇒ 2x < 8
⇒ x < 4
⇒ x ∈ (- ∞ , 4)
⇒ f is strictly decreasing in (- ∞ , 4)
Let f be a real valued function such that f: I → R.
Let x1 ∈ I and h > 0
Consider (x1 - h, x1 + h)
(1)f is said to be increasing at x1, if f is increasing in (x1 - h, x1 + h)
(2)f is said to be strictly increasing at x1, if f is strictly increasing in (x1 - h, x1 + h)
(3)f is said to be decreasing at x1, if f is decreasing in (x1 - h, x1 + h)
(4)f is said to be strictly decreasing at x1, if f is strictly decreasing in (x1 - h, x1 + h)
Theorem: Let f be continuous on [a, b] and differentiable on (a, b) Then:
(i) f is increasing in [a, b], if f ' (x) ≥ 0, ∀ x ∈ (a, b)
(ii) f is decreasing in [a, b], if f ' (x) ≤ 0, ∀ x ∈ (a, b)
(iii) f is a constant function in [a, b] if f ' (x) = 0, ∀ x ∈ (a, b)
Proof:
Let x1, x2 ∈ [a, b] such that x1 < x2
By the mean value theorem, there exists c ∈ (x1, x2) such that f(c) = ....(1)
(i)Let f(c) ≥ 0
⇒ ≥ 0
⇒ ≥ 0 (Since x1 < x2)
⇒
Or
⇒ f is an increasing function.
(ii) Let f(c) ≤ 0
⇒ ≤ 0
(Since x1 < x2)
⇒
Or
⇒ f is a decreasing function.
(iii)Let f(c) = 0
⇒ = 0
⇒ = 0
⇒ , ∀ x1, x2 ∈ [a, b]
⇒
Some results:
(I)f is strictly increasing in (a, b) if f ' (x) > 0, ∀ x ∈ [a, b]
(II)f is strictly decreasing in (a, b) if f ' (x) < 0, ∀ x ∈ [a, b]
(III)f is increasing or decreasing on R if it is increasing or decreasing in every interval of R
Function f is increasing in [a, b], if f ' (x) ≥ 0, ∀ x ∈ (a, b).
Function f is decreasing in [a, b], if f ' (x) ≤ 0, ∀ x ∈ (a, b).
Function f is a constant function in [a, b] if f ' (x) = 0, ∀ x ∈ (a, b).
Example :
Show that the function f(x) = x3 – 6x2 + 15x + 3 is strictly increasing on the set of real numbers.
Sol :
Given, f(x) = x3 – 6x2 + 15x + 3
Differentiating with respect to x, we get
f(x) = 3x2 - 12x + 15
= 3(x2 - 4x + 5)
= 3(x2 - 4x + 4 + 1)
= 3((x - 2)2 + 1)
Since (x - 2)2 ≥ 1, ∀ x ∈ R
⇒ 3((x - 2)2 + 1) > 0, ∀ x ∈ R
⇒ f ' (x) > 0, ∀ x ∈ R
⇒ f(x) = x3 - 6x2 + 15x + 3 is strictly increasing on the set of real numbers.
Example :
Find the intervals in which the function, f, given by f(x) = x2 - 8x + 5 is:
(a) Strictly increasing
(b) Strictly decreasing
Sol :
Given f(x) = x2
Differentiating with respect to x, we get f(x) = 2x - 8
a) f is strictly increasing if f(x)
⇒ 2x - 8 > 0
⇒ 2x > 8
⇒ x > 4
⇒ x ∈ (4, ∞)
⇒ f is strictly increasing in (4, ∞)
b) f is strictly decreasing if f(x) < 0
⇒ 2x - 8 < 0
⇒ 2x < 8
⇒ x < 4
⇒ x ∈ (- ∞ , 4)
⇒ f is strictly decreasing in (- ∞ , 4)