Advertisements
Advertisements
Question
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.
Advertisements
Solution
Here,
f(x) = `(4 sin x - 2x - x cos x)/(2 + cos x)`
`= (4 sin x)/(2 + cos x) - x`
∴ f(x) = `((2 + cos x)4 cos x - 4 sin x (- sin x))/(2 + cos x)^2 - 1`
`= (8 cos x + 4 cos^2 x + 4 sin^2 x)/(2 + cos x)^2 - 1`
`= (8 cos x + 4 - (2 + cos x)^2)/(2 + cos x)`
`= (4 cos x - cos^2 x)/((2 + cos x)^2)`
`= (cos x (4 - cos x))/(2 + cos x)^2`
because – 1 ≤ cos x ≤ 1
⇒ 4 - cos x > 0 and (2 + cos x)2 > 0
∴ f(x) > 0 or < 0 such that cos x > 0 or cos x < 0 respectively
∴ f(x) is increasing when 0 < x < `pi/2, (3pi)/2 < x < 2 pi`
And f(x) is decreasing when `pi/2 < pi < (3pi)/2`.
APPEARS IN
RELATED QUESTIONS
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the logarithmic function is strictly increasing on (0, ∞).
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?
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(x) = 2x3 + 9x2 + 12x + 20 ?
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(x) = x4 − 4x ?
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 increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3) ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
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) = x|x|, x \[\in\] R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Find the values of 'a' for which the function f(x) = sin x − ax + 4 is increasing function on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Function f(x) = loga x is increasing on R, if
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
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 values of x for which the following functions are strictly decreasing : f(x) = x3 – 9x2 + 24x + 12
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
The function f(x) = 9 - x5 - x7 is decreasing for
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
y = x(x – 3)2 decreases for the values of x given by : ______.
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
The interval in which `y = x^2e^(-x)` is increasing with respect to `x` is
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
