Advertisements
Advertisements
Question
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
Options
(0,1)
`(pi/2, pi)`
`(0, pi/2)`
None of these
Advertisements
Solution
None of these
Explanation:
Given f(x) = x100 + sin x - 1,
f'(x) = 100 x99 + cos x
(a) Interval 0 < x < 1, 0 < 100 x99 < 100
And cos x = + positive
`therefore` f'(x) = + positive
Hence, the function f is increasing.
(b) Interval is `pi/2 < "x" < pi`
`therefore` f'(x) = 100 x99 + cos x = + positive
Hence, the function f is increasing.
(c) Interval is, `0 < "x" < pi/2`
Here, 100 x99 and cos x are both positive.
`therefore` f'(x) = + ve
Hence, the function f is increasing.
APPEARS IN
RELATED QUESTIONS
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 − 9x − x2 ?
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 + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
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) = sin x is an increasing function on (−π/2, π/2) ?
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) ?
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
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 f(x) = x9 + 4x7 + 11 is an increasing function for all x ∈ R ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Function f(x) = ax is increasing on R, if
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Show that f(x) = x – cos x is increasing for all x.
Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.
The values of a for which the function f(x) = sinx – ax + b increases on R are ______.
The function f (x) = x2, for all real x, is ____________.
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
Let f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
