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
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
The interval in which y = x2 e–x is increasing is ______.
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = 3 x^4 - 4 x^3 - 12 x^2 + 5\] ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
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 following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
The function f(x) = x2 e−x is monotonic increasing when
Function f(x) = cos x − 2 λ x is monotonic decreasing when
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
Function f(x) = ax is increasing on R, if
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
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) = x – cos x is increasing for all x.
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
