Advertisements
Advertisements
प्रश्न
Find the values of x for which f(x) = `x/(x^2 + 1)` is (a) strictly increasing (b) decreasing.
Advertisements
उत्तर
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
संबंधित प्रश्न
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 following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Prove that the logarithmic function is strictly increasing on (0, ∞).
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
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.
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Show that the function f given by f(x) = 10x is increasing for all x ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
The function f(x) = cot−1 x + x increases in the interval
The function f(x) = xx decreases on the interval
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
Function f(x) = cos x − 2 λ x is monotonic decreasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = | x | − | x − 1 | is monotonically increasing when
Function f(x) = loga x is increasing on R, if
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
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
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Show that f(x) = x – cos x is increasing for all x.
The slope of tangent at any point (a, b) is also called as ______.
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
The function f(x) = 9 - x5 - x7 is decreasing for
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
The function `1/(1 + x^2)` is increasing in the interval ______
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f (x) = 2 – 3 x is ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
Function given by f(x) = sin x is strictly increasing in.
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
If f(x) = x + cosx – a then ______.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
If \[f'(x)>0\] throughout an interval, then \[f\] is
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
In the procedure for finding intervals, which conclusion is correct when \[f'(x)<0\]?
