Advertisements
Advertisements
प्रश्न
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Advertisements
उत्तर
\[f\left( x \right) = \tan^{- 1} \left( \sin x + \cos x \right)\]
\[f'\left( x \right) = \frac{1}{1 + \left( \sin x + \cos x \right)^2}\left( \cos x - \sin x \right)\]
\[ = \frac{1}{1 + 1 + 2 \sin x \cos x}\left( \cos x - \sin x \right)\]
\[ = \frac{\left( \cos x - \sin x \right)}{2 + \sin 2x}\]
\[\text { Here },\]
\[\frac{\pi}{4} < x < \frac{\pi}{2}\]
\[ \Rightarrow \frac{\pi}{2} < 2x < \pi\]
\[ \Rightarrow \sin 2x > 0\]
\[ \Rightarrow 2 + \sin 2x > 0 . . . \left( 1 \right)\]
\[\text { Also,} \]
\[\frac{\pi}{4} < x < \frac{\pi}{2}\]
\[\cos x < \sin x\]
\[ \Rightarrow \cos x - \sin x < 0 . . . \left( 2 \right)\]
\[f'\left( x \right) = \frac{\left( \cos x - \sin x \right)}{2 + \sin 2x} < 0, \forall x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \left[ \text { From eqs } . (1) \text { and } (2) \right]\]
\[\text { So },f\left( x \right)\text { is decreasing on }\left( \frac{\pi}{4}, \frac{\pi}{2} \right).\]
APPEARS IN
संबंधित प्रश्न
The interval in which y = x2 e–x is increasing is ______.
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
Show that f(x) = x − sin x is increasing for all x ∈ R ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
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. \[\left[ \text{ Use } \pi = \frac{22}{7} \right]\]
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
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).
Prove that the function f : N → N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Test whether the following functions are increasing or decreasing: f(x) = `x-(1)/x`, x ∈ R, x ≠ 0.
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Find the value of x such that f(x) is decreasing function.
f(x) = x4 − 2x3 + 1
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
The function f(x) = sin x + 2x is ______
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
In `(0, pi/2),` the function f (x) = `"x"/"sin x"` is ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
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)`.
If f(x) = x + cosx – a then ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
The function \[f(x)=x^3-3x^2+4x\], \[x\in\mathbf{R}\], is
