Advertisements
Advertisements
Question
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
Options
increasing on (0, π/2)
decreasing on (0, π/2)
increasing on (0, π/4) and decreasing on (π/4, π/2)
none of these
Advertisements
Solution
increasing on (0, \[\pi\]/2)
\[\text { Given:}g\left( x \right) \text { is increasing on }\left( 0, \frac{\pi}{2} \right). \text { Then, }\]
\[ x_1 < x_2 , \forall x_1 , x_2 \in \left( 0, \frac{\pi}{2} \right)\]
\[ \Rightarrow g\left( x_1 \right) < g\left( x_2 \right)\]
\[ {\text { Taking } tan}^{- 1} \text { on both the sides, we get } \]
\[ \tan^{- 1} \left( g\left( x_1 \right) \right) < \tan^{- 1} \left( g\left( x_2 \right) \right)\]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, \frac{\pi}{2} \right)\]
\[\text { So,}f\left( x \right)\text { is increasing on }\left( 0, \frac{\pi}{2} \right).\]
APPEARS IN
RELATED QUESTIONS
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
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:
x2 + 2x − 5
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
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.
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
The interval in which y = x2 e–x is increasing is ______.
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
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) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = sin x is an increasing function on (−π/2, π/2) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
Every invertible function is
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
For every value of x, the function f(x) = `1/7^x` is ______
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
A function f is said to be increasing at a point c if ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f(x) = x3 + 3x is increasing in interval ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
