Advertisements
Advertisements
प्रश्न
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Advertisements
उत्तर
\[f\left( x \right) = x^2 - x \sin x\]
\[f'\left( x \right) = 2x - x \cos x - \sin x\]
\[\text { Here,} \]
\[0 < x < \frac{\pi}{2}\]
\[ \Rightarrow 0 < \sin x < 1 \text { and }0 < \cos x < 1\]
\[ \Rightarrow 2x - x \cos x - \sin x > 0\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( 0, \frac{\pi}{2} \right)\]
\[\text { So },f\left( x \right)\text { is increasing on}\left( 0, \frac{\pi}{2} \right).\]
APPEARS IN
संबंधित प्रश्न
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 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 \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
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] ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
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
The function f(x) = x2 e−x is monotonic increasing when
The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
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] ______
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
The function `1/(1 + x^2)` is increasing in the interval ______
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Which of the following functions is decreasing on `(0, pi/2)`?
The function `"f"("x") = "x"/"logx"` increases on the interval
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
y = log x satisfies for x > 1, the inequality ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
