Advertisements
Advertisements
Question
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
Advertisements
Solution
Given `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi`
⇒ `f("x") = ((2+cos"x")4 cos "x" + 4 sin^2 "x")/(2+ cos"x")^2 - 1`
⇒ `f("x") = (8 cos "x" + 4(sin^2 "x" + cos^2 "x")-4-cos^2 "x"-4cos"x")/((2+ cos"x"))`
⇒ `f("x") = (4cos "x"- cos^2 "x")/((2 + cos "x")^2`
For critical points, `f("x") = [(4- cos"x")/((2+ cos "x")^2]] cos "x" = 0`
f(x) is strictly increasing for f'(x) > 0
i.e., cos x > 0 ⇒ x ∈ `[0, pi/2), ∪ (3pi/2, 2pi ]`
and f(x) is strictly decreasing for f'(x) < 0
i.e., cos x < 0 ⇒ x ∈
| Interval | Sign of f (x) | f (x) is strictly |
| (0,π/2) | Positive | Increasing |
| (π/2, 3π/2) | Negative | Decreasing |
| (3π/2,2π) | Positive | Increasing |
APPEARS IN
RELATED QUESTIONS
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
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) = x3 − 6x2 − 36x + 2 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
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) = sin x is an increasing function on (−π/2, 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 the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
Let f(x) = x3 − 6x2 + 15x + 3. Then,
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
Find MPC ( Marginal propensity to Consume ) and APC ( Average Propensity to Consume ) if the expenditure Ec of a person with income I is given as Ec = ( 0.0003 ) I2 + ( 0.075 ) I when I = 1000.
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The function f(x) = x3 - 3x is ______.
Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`
The function f (x) = x2, for all real x, 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:
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
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))` = ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The function f(x) = x3 + 3x is increasing in interval ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
