Advertisements
Advertisements
प्रश्न
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Advertisements
उत्तर
\[f\left( x \right) = \sin \left( 2x + \frac{\pi}{4} \right)\]
\[f'\left( x \right) = 2 \cos \left( 2x + \frac{\pi}{4} \right)\]
\[\text { Here, } \]
\[\frac{3\pi}{8} < x < \frac{5\pi}{8}\]
\[ \Rightarrow \frac{3\pi}{4} < 2x < \frac{5\pi}{4}\]
\[ \Rightarrow \pi < 2x + \frac{\pi}{4} < \frac{3\pi}{2}\]
\[ \Rightarrow \ cos \left( 2x + \frac{\pi}{4} \right) < 0 \left[ \because \text { Cos function is negative inthird quadrant } \right]\]
\[ \Rightarrow 2 \cos \left( 2x + \frac{\pi}{4} \right) < 0\]
\[ \Rightarrow f'\left( x \right) < 0, \forall x \in \left( \frac{3\pi}{8}, \frac{5\pi}{8} \right)\]
\[\text { So },f\left( x \right) \text { is decreasing on }\left( \frac{3\pi}{8}, \frac{5\pi}{8} \right).\]
APPEARS IN
संबंधित प्रश्न
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
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) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
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 \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
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) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Let f defined on [0, 1] be twice differentiable such that | f (x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1] ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
Let f(x) = x3 − 6x2 + 15x + 3. Then,
Function f(x) = cos x − 2 λ x is monotonic decreasing when
Every invertible function is
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
The function f(x) = tanx – x ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
Let h(x) = f(x) - [f(x)]2 + [f(x)]3 for every real number x. Then ____________.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) 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))` = ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
