Advertisements
Advertisements
Question
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
Advertisements
Solution
\[ f\left( x \right) = \sin x + \left| \sin x \right|, 0 < x \leq 2\pi\]
\[\text { Case I: When x }\in \left( 0, \pi \right)\]
\[f\left( x \right) = \sin x + \sin x = 2\sin x\]
\[ \Rightarrow f'\left( x \right) = 2\cos x\]
\[\text { As,} \cos x > 0 \text { for } x \in \left( 0, \frac{\pi}{2} \right) \text { and }\cos x < 0 \text { for } x \in \left( \frac{\pi}{2}, \pi \right)\]
\[\text { So,} f'\left( x \right) > 0\text { for} x \in \left( 0, \frac{\pi}{2} \right)\text{ and } f'\left( x \right) < 0 \text { for }x \in \left( \frac{\pi}{2}, \pi \right)\]
\[ \therefore f\left( x \right)\text { is increaing on} \left( 0, \frac{\pi}{2} \right) \text { and } f\left( x \right) \text { is decreasing on } \left( \frac{\pi}{2}, \pi \right) . \]
\[\text { Case II: When x } \in \left( \pi, 2\pi \right)\]
\[f\left( x \right) = \sin x - \sin x = 0\]
\[ \Rightarrow f'\left( x \right) = 0\]
\[\text { So,} f\left( x \right) \text { is neither increaing nor decreasing on } \left( \pi, 2\pi \right) . \]
APPEARS IN
RELATED QUESTIONS
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
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).
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
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 − 9x2 + 12x − 5 ?
Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
Let \[f\left( x \right) = \tan^{- 1} \left( g\left( x \right) \right),\],where g (x) is monotonically increasing for 0 < x < \[\frac{\pi}{2} .\] Then, f(x) is
The function f(x) = x2 e−x is monotonic increasing when
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Show that f(x) = x – cos x is increasing for all x.
The slope of tangent at any point (a, b) is also called as ______.
State whether the following statement is True or False:
If the function f(x) = x2 + 2x – 5 is an increasing function, then x < – 1
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Find the interval/s in which the function f : R `rightarrow` R defined by f(x) = xex, is increasing.
The function f(x) = sin4x + cos4x is an increasing function if ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
Let \[f\] be continuous on \[[a,b]\] and differentiable on \[(a,b)\]. If \[f'(x)\leq0\] for every \[x\in(a,b)\], what follows?
Which form shows that \[f'(x)=3x^2-6x+4\] is positive for every \[x\in\mathbf{R}\]?
