Advertisements
Advertisements
Question
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
Options
strictly increasing
strictly decreasing
neither increasing nor decreasing
none of these
Advertisements
Solution
strictly increasing
\[f\left( x \right) = \frac{x}{1 + \left| x \right|}\]
\[\text { Case 1: When }x > 0, \left| x \right| = x\]
\[f\left( x \right) = \frac{x}{1 + \left| x \right|}\]
\[ = \frac{x}{1 + x}\]
\[ \Rightarrow f'\left( x \right) = \frac{\left( 1 + x \right)1 - x\left( 1 \right)}{\left( 1 + x \right)^2}\]
\[ = \frac{1}{\left( 1 + x \right)^2} > 0, \forall x \in R\]
\[\text { So,f }\left( x \right) \text { is strictly increasing when }x> 0.\]
\[\text { Case 2: When }x < 0, \left| x \right| = - x\]
\[f\left( x \right) = \frac{x}{1 + \left| x \right|}\]
\[ = \frac{x}{1 - x}\]
\[ \Rightarrow f'\left( x \right) = \frac{\left( 1 - x \right)1 - x\left( - 1 \right)}{\left( 1 - x \right)^2}\]
\[ = \frac{1}{\left( 1 - x \right)^2} > 0, \forall x \in R\]
\[\text { So,f }\left( x \right) \text { is strictly increasing when }x <0.\]
\[\text { Thus,f }\left( x \right) \text { is strictly increasing on R } . \]
APPEARS IN
RELATED QUESTIONS
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
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)
Prove that the logarithmic function is strictly increasing on (0, ∞).
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) 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 f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
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) = x2 + 2x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
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 ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
The function f(x) = xx decreases on the interval
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
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 values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
The slope of tangent at any point (a, b) is also called as ______.
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = tanx – x ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
In case of decreasing functions, slope of tangent and hence derivative is ____________.
The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
