Advertisements
Advertisements
Question
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Advertisements
Solution
\[f\left( x \right) = x^2 - x \sin x\]
\[f'\left( x \right) = 2x - x \cos x - \sin x\]
\[\text { Here,} \]
\[0 < x < \frac{\pi}{2}\]
\[ \Rightarrow 0 < \sin x < 1 \text { and }0 < \cos x < 1\]
\[ \Rightarrow 2x - x \cos x - \sin x > 0\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( 0, \frac{\pi}{2} \right)\]
\[\text { So },f\left( x \right)\text { is increasing on}\left( 0, \frac{\pi}{2} \right).\]
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
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) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 + 9x2 + 12x + 20 ?
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) = tan x is an increasing function on (−π/2, π/2) ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4)?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Every invertible function is
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
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.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Show that f(x) = x – cos x is increasing for all x.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
For every value of x, the function f(x) = `1/"a"^x`, a > 0 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 ______
The function `1/(1 + x^2)` is increasing in the interval ______
y = x(x – 3)2 decreases for the values of x given by : ______.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
