Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
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 \[f\left( x \right) = \frac{3}{10} x^4 - \frac{4}{5} x^3 - 3 x^2 + \frac{36}{5}x + 11\] ?
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) = e2x is increasing on R.
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
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 ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing ?
The function f(x) = cot−1 x + x increases in the interval
Let f(x) = x3 − 6x2 + 15x + 3. Then,
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = loga x is increasing on R, if
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
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 price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
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 func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Choose the correct alternative.
The function f(x) = x3 - 3x2 + 3x - 100, x ∈ R is
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
The slope of tangent at any point (a, b) is also called as ______.
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The function f(x) = x3 - 3x is ______.
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
y = x(x – 3)2 decreases for the values of x given by : ______.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
Let f(x) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
