Advertisements
Advertisements
प्रश्न
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Advertisements
उत्तर
\[f\left( x \right) = \sin x - \cos x, x \in \left( 0, 2\pi \right)\]
\[f'\left( x \right) = \cos x + \sin x\]
\[\text { For f(x) to be increasin, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \cos x + \sin x > 0\]
\[ \Rightarrow \sin x > - \cos x\]
\[ \Rightarrow \tan x > - 1\]
\[ \Rightarrow x \in \left( 0, \frac{3\pi}{4} \right) \cup \left( \frac{7\pi}{4}, 2\pi \right)\]
\[\text { So,f(x)is increasing on } \left( 0, \frac{3\pi}{4} \right) \cup \left( \frac{7\pi}{4}, 2\pi \right) . \]
\[\text { For f(x) to be decreasing we must have},\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \cos x + \sin x < 0\]
\[ \Rightarrow \sin x < - \cos x\]
\[ \Rightarrow \tan x < - 1\]
\[ \Rightarrow x \in \left( \frac{3\pi}{4}, \frac{7\pi}{4} \right)\]
\[\text { So,f(x)is decreasing on }\left( \frac{3\pi}{4}, \frac{7\pi}{4} \right).\]
APPEARS IN
संबंधित प्रश्न
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 function f given by f(x) = 2x3 − 3x2 − 36x + 7 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.
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
The interval in which y = x2 e–x is increasing is ______.
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
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) = 2x3 − 15x2 + 36x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
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 f(x) = tan−1 x − x is a decreasing function on R ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Every invertible function is
Function f(x) = ax is increasing on R, if
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Prove that y = `(4sinθ)/(2 + cosθ) - θ` is an increasing function if `θ ∈[0, pi/2]`
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Choose the correct alternative:
The function f(x) = x3 – 3x2 + 3x – 100, x ∈ R is
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
The function `1/(1 + x^2)` is increasing in the interval ______
Show that f(x) = tan–1(sinx + cosx) is an increasing function in `(0, pi/4)`
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
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 ______.
y = log x satisfies for x > 1, the inequality ______.
