Advertisements
Advertisements
Question
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Advertisements
Solution
\[\text { Here,} \]
\[f\left( x \right) = \sin x\]
\[\text { Domain of sin x is }\left( 0, \pi \right).\]
\[f'\left( x \right) = \cos x\]
\[\text { For x } \in \left( 0, \frac{\pi}{2} \right), \cos x > 0 \left[ \because \cos x\text { is positive in first quadrant} \right]\]
\[f'\left( x \right) > 0\]
\[\text { So,f(x)is increasing for
}\left( 0, \frac{\pi}{2} \right) . \]
\[\text { For x} \in \left( \frac{\pi}{2}, \pi \right), \cos x < 0 \left[ \because \cos x\text { is negative in second quadrant } \right]\]
\[\text { So,f(x)is decreasing for }\left( \frac{\pi}{2}, \pi \right).\]
\[\text { Since }f(x)\text { is increasing on } \left( 0, \frac{\pi}{2} \right) \text { and decreasing on}\left( \frac{\pi}{2}, \pi \right), f\left( x \right) \text { is neither decreasing nor increasing on }\left( 0, \pi \right).\]
APPEARS IN
RELATED QUESTIONS
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) decreasing.
Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 8 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Show that f(x) = e2x is increasing on R.
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing 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 'a' for which f(x) = x + cos x + ax + b is increasing on R ?
The function f(x) = xx decreases on the interval
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Every invertible function is
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Show that f(x) = x – cos x is increasing for all x.
Let f(x) = x3 − 6x2 + 9𝑥 + 18, then f(x) is strictly decreasing in ______
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
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] ______
Let f(x) = x3 + 9x2 + 33x + 13, then f(x) is ______.
The function f (x) = x2, for all real x, is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
Let f(x) = tan–1`phi`(x), where `phi`(x) is monotonically increasing for `0 < x < π/2`. Then f(x) is ______.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
y = log x satisfies for x > 1, the inequality ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
A function f is said to be increasing at a point c if ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
