Advertisements
Advertisements
Question
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
Options
increasing
decreasing
constant
none of these
Advertisements
Solution
\[f(x) = \frac{- x}{2} + \sin x\text { defined on } \left[ \frac{- \pi}{3}, \frac{\pi}{3} \right]\]
\[ \therefore f'(x) = \frac{- 1}{2} + \cos x \]
\[ \Rightarrow f'(x) \geqslant 0 \forall x \in \left[ \frac{- \pi}{3}, \frac{\pi}{3} \right]\]
\[\left[ \because \text { for } x \in \left[ \frac{- \pi}{3}, \frac{\pi}{3} \right] , \cos x \geqslant \frac{1}{2} \right]\]
Hence, the given function is increasing .
APPEARS IN
RELATED QUESTIONS
Price P for demand D is given as P = 183 +120D - 3D2 Find D for which the price is increasing
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Find the interval in which the following function are increasing or decreasing f(x) = 5x3 − 15x2 − 120x + 3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
Find the interval in which the following function are increasing or decreasing f(x) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Find the values of b for which the function f(x) = sin x − bx + c is a decreasing function on R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = cot−1 x + x increases in the interval
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
The function f(x) = x9 + 3x7 + 64 is increasing on
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
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).
Find the intervals in which function f given by f(x) = 4x3 - 6x2 - 72x + 30 is (a) strictly increasing, (b) strictly decresing .
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 values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
show that f(x) = `3x + (1)/(3x)` is increasing in `(1/3, 1)` and decreasing in `(1/9, 1/3)`.
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
f(x) = `{{:(0"," x = 0 ), (x - 3"," x > 0):}` The function f(x) is ______
The function f(x) = tanx – x ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f(x) = x2 – 2x is increasing in the interval ____________.
Let `"f (x) = x – cos x, x" in "R"`, then f is ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
