Advertisements
Advertisements
प्रश्न
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Advertisements
उत्तर
\[f\left( x \right) = x - \sin x\]
\[f'\left( x \right) = 1 - \cos x\]
\[\text { For f(x) to be increasing, we must have}\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 1 - \cos x > 0\]
\[ \Rightarrow f'(x) \geqslant 0 \text { for all } x \in R \left[ \because Cos x \leqslant 1 \right]\]
\[\text { So, f(x) is increasing for all } x \in R . \]
APPEARS IN
संबंधित प्रश्न
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
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)`
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
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(x) = x4 − 4x3 + 4x2 + 15 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Find `dy/dx,if e^x+e^y=e^(x-y)`
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.
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Solve the following:
Find the intervals on which the function f(x) = `x/logx` is increasing and decreasing.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
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
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
Which of the following functions is decreasing on `(0, pi/2)`?
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
Show that function f(x) = tan x is increasing in `(0, π/2)`.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
The function f(x) = x3 + 3x is increasing in interval ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
If \[f'(x)>0\] throughout an interval, then \[f\] is
As one moves from left to right on a graph, what does an increase in the \[y\]-values indicate?
After the domain has been divided into intervals, which quantity must be determined in each interval?
For \[f(x)=4x^3-6x^2-72x+30\], which factorization of \[f'(x)\] is correct?
