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.
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
The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
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 ?
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 − 15x2 + 36x + 1 ?
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 \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = x2 − x sin x is an increasing function on (0, π/2) ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Let f(x) = x3 − 6x2 + 15x + 3. Then,
If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function f(x) = x9 + 3x7 + 64 is increasing on
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 values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
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.
Choose the correct option from the given alternatives :
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly decreasing in ______.
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Find the values of x for which the function f(x) = x3 – 6x2 – 36x + 7 is strictly increasing
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
The area of the square increases at the rate of 0.5 cm2/sec. The rate at which its perimeter is increasing when the side of the square is 10 cm long is ______.
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
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: ____________.
If f(x) = sin x – cos x, then interval in which function is decreasing in 0 ≤ x ≤ 2 π, is:
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The function f(x) = x3 + 3x is increasing in interval ______.
