Advertisements
Advertisements
Question
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Advertisements
Solution
f(x) = `x/(x^2 + 1)`
∴ f'(x) = `d/dx(x/(x^2 + 1))`
= `((x^2 + 1).d/dx(x) - xd/dx(x^2 + 1))/(x^2 + 1)^2`
= `((x^2 + 1)(1) - x(2x + 0))/(x^2 + 1)^2`
= `(x^2 + 1 - 2x^2)/(x^2 + 1)^2`
= `(1 - x^2)/(x^2 + 1)^2`
(a) f is strictly increasing if f'(x) > 0
i.e. if `(1 - x^2)/(x^2 + 1)^2 > 0`
i.e. if 1 – x2 > 0 ...[∵ (x2 + 1)2 > 0]
i.e. if 1 > x2
i.e. if x2 < 1
i.e. if – 1 < x < 1
∴ f is strictly increasing if – 1 < x < 1
(b) f is strictly decreasing if f'(x) < 0
i.e. if `(1 - x^2)/(x^2 + 1)^2 < 0`
i.e. if 1 – x2 < 0 ...[∵ (x2 + 1)2 > 0]
i.e. if 1 < x2
i.e. if x2 > 1
i.e. if x > 1 or x < – 1
∴ f is strictly decreasing if x < – 1 or x > 1
i.e. `x ∈( - oo, - 1) ∪ (1, oo)`.
APPEARS IN
RELATED QUESTIONS
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \left\{ x(x - 2) \right\}^2\] ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
Function f(x) = cos x − 2 λ x is monotonic decreasing when
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the values of x for which Revenue is increasing.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = tanx – x ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
Function given by f(x) = sin x is strictly increasing in.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
A function f is said to be increasing at a point c if ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
