Advertisements
Advertisements
प्रश्न
Find the interval in which f(x) is increasing or decreasing f(x) = sinx + |sin x|, 0 < x \[\leq 2\pi\] ?
Advertisements
उत्तर
\[ f\left( x \right) = \sin x + \left| \sin x \right|, 0 < x \leq 2\pi\]
\[\text { Case I: When x }\in \left( 0, \pi \right)\]
\[f\left( x \right) = \sin x + \sin x = 2\sin x\]
\[ \Rightarrow f'\left( x \right) = 2\cos x\]
\[\text { As,} \cos x > 0 \text { for } x \in \left( 0, \frac{\pi}{2} \right) \text { and }\cos x < 0 \text { for } x \in \left( \frac{\pi}{2}, \pi \right)\]
\[\text { So,} f'\left( x \right) > 0\text { for} x \in \left( 0, \frac{\pi}{2} \right)\text{ and } f'\left( x \right) < 0 \text { for }x \in \left( \frac{\pi}{2}, \pi \right)\]
\[ \therefore f\left( x \right)\text { is increaing on} \left( 0, \frac{\pi}{2} \right) \text { and } f\left( x \right) \text { is decreasing on } \left( \frac{\pi}{2}, \pi \right) . \]
\[\text { Case II: When x } \in \left( \pi, 2\pi \right)\]
\[f\left( x \right) = \sin x - \sin x = 0\]
\[ \Rightarrow f'\left( x \right) = 0\]
\[\text { So,} f\left( x \right) \text { is neither increaing nor decreasing on } \left( \pi, 2\pi \right) . \]
APPEARS IN
संबंधित प्रश्न
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
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) = x8 + 6x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
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) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
The function f(x) = cot−1 x + x increases in the interval
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
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 func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
2x3 - 6x + 5 is an increasing function, if ____________.
The function f: N → N, where
f(n) = `{{:(1/2(n + 1), "If n is sold"),(1/2n, "if n is even"):}` is
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
A function f is said to be increasing at a point c if ______.
The function f(x) = x3 + 3x is increasing in interval ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
