Advertisements
Advertisements
Question
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Advertisements
Solution
\[f\left( x \right) = \tan^{- 1} \left( \sin x + \cos x \right)\]
\[f'\left( x \right) = \frac{1}{1 + \left( \sin x + \cos x \right)^2}\left( \cos x - \sin x \right)\]
\[ = \frac{1}{1 + 1 + 2 \sin x \cos x}\left( \cos x - \sin x \right)\]
\[ = \frac{\left( \cos x - \sin x \right)}{2 + \sin 2x}\]
\[\text { Here },\]
\[\frac{\pi}{4} < x < \frac{\pi}{2}\]
\[ \Rightarrow \frac{\pi}{2} < 2x < \pi\]
\[ \Rightarrow \sin 2x > 0\]
\[ \Rightarrow 2 + \sin 2x > 0 . . . \left( 1 \right)\]
\[\text { Also,} \]
\[\frac{\pi}{4} < x < \frac{\pi}{2}\]
\[\cos x < \sin x\]
\[ \Rightarrow \cos x - \sin x < 0 . . . \left( 2 \right)\]
\[f'\left( x \right) = \frac{\left( \cos x - \sin x \right)}{2 + \sin 2x} < 0, \forall x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \left[ \text { From eqs } . (1) \text { and } (2) \right]\]
\[\text { So },f\left( x \right)\text { is decreasing on }\left( \frac{\pi}{4}, \frac{\pi}{2} \right).\]
APPEARS IN
RELATED QUESTIONS
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
The interval in which y = x2 e–x is increasing is ______.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on 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 intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0 ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R ?
Let f(x) = x3 − 6x2 + 15x + 3. Then,
Function f(x) = cos x − 2 λ x is monotonic decreasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function f(x) = x9 + 3x7 + 64 is increasing on
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 value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The function f(x) = x3 - 3x is ______.
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The function f(x) = tanx – x ______.
The function f(x) = tan-1 x is ____________.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
The critical points \[x=-2\] and \[x=3\] divide the real line into which three intervals?
For \[f(x)=4x^3-6x^2-72x+30\], on which intervals is \[f\] increasing?
