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,
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
Show that the function given by f(x) = 3x + 17 is strictly increasing on R.
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the intervals in which the following functions are strictly increasing or decreasing:
6 − 9x − x2
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .
Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x2 ?
Find the interval in which the following function are increasing or decreasing f(x) = −2x3 − 9x2 − 12x + 1 ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
Show that f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = cos x − 2 λ x is monotonic decreasing when
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
The function f(x) = x9 + 3x7 + 64 is increasing on
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Find the value of x, such that f(x) is increasing function.
f(x) = x2 + 2x - 5
Show that the function f(x) = x3 + 10x + 7 for x ∈ R is strictly increasing
The slope of tangent at any point (a, b) is also called as ______.
The total cost function for production of articles is given as C = 100 + 600x – 3x2, then the values of x for which the total cost is decreasing is ______
The function f(x) = sin x + 2x is ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
The function f(x) = `|x - 1|/x^2` is monotonically decreasing on ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
If \[f'(x)<0\] throughout an interval, then \[f\] is
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
As one moves from left to right on a graph, what do constant \[y\]-values indicate?
For \[f(x)=4x^3-6x^2-72x+30\], which factorization of \[f'(x)\] is correct?
Which statement about \[f'(x)=0\] and constant behaviour is correct?
