Advertisements
Advertisements
प्रश्न
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
विकल्प
(0,1)
`(pi/2, pi)`
`(0, pi/2)`
None of these
Advertisements
उत्तर
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
संबंधित प्रश्न
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) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
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) = (x − 1) (x − 2)2 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Show that f(x) = e2x is increasing on R.
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?
Find the value(s) of a for which f(x) = x3 − ax is an increasing function on R ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
Function f(x) = x3 − 27x + 5 is monotonically increasing when ______.
Function f(x) = | x | − | x − 1 | is monotonically increasing when
The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
The price P for demand D is given as P = 183 + 120 D – 3D2.
Find D for which the price is increasing.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
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
Test whether the following function f(x) = 2 – 3x + 3x2 – x3, x ∈ R is increasing or decreasing
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
The slope of tangent at any point (a, b) is also called as ______.
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing function
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
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) = tanx – x ______.
The function f (x) = 2 – 3 x is ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
2x3 - 6x + 5 is an increasing function, if ____________.
Which of the following graph represent the strictly increasing function.
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 ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The function f(x) = tan–1(sin x + cos x) is an increasing function in ______.
If f(x) = `x/(x^2 + 1)` is increasing function then the value of x lies in ______.
Read the following passage:
|
The use of electric vehicles will curb air pollution in the long run. V(t) = `1/5 t^3 - 5/2 t^2 + 25t - 2` where t represents the time and t = 1, 2, 3, ...... corresponds to years 2001, 2002, 2003, ...... respectively. |
Based on the above information, answer the following questions:
- Can the above function be used to estimate number of vehicles in the year 2000? Justify. (2)
- Prove that the function V(t) is an increasing function. (2)
