Advertisements
Advertisements
प्रश्न
Prove that the function given by f (x) = x3 – 3x2 + 3x – 100 is increasing in R.
Advertisements
उत्तर
It is known that- f (x) = x3 - 3x2 + 3x - 100
`therefore` f'(x) = 3x2 - 6x + 3
= 3 (x2 - 2x + 1)
= 3 (x - 1)2 ≥ 0 for all `x in R`
= 3(x - 1)2 > 0
∀ x ∈ R, f''(x) = positive
Hence, the function f is increasing.
APPEARS IN
संबंधित प्रश्न
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
On which of the following intervals is the function f given byf(x) = x100 + sin x –1 strictly decreasing?
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 f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
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 intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π) ?
Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all x ∈ R ?
Show that f(x) = cos2 x is a decreasing function on (0, π/2) ?
State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6] ?
Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2) ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Find `dy/dx,if e^x+e^y=e^(x-y)`
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Solve the following : Find the intervals on which the function y = xx, (x > 0) is increasing and decreasing.
Test whether the following function is increasing or decreasing.
f(x) = `7/"x" - 3`, x ∈ R, x ≠ 0
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 + 36x + 1
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
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
The function f(x) = `x - 1/x`, x ∈ R, x ≠ 0 is increasing
The function f(x) = 9 - x5 - x7 is decreasing for
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
The function f (x) = 2 – 3 x is ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.
The interval in which the function f(x) = `(4x^2 + 1)/x` is decreasing is ______.
Which form shows that \[f'(x)=3x^2-6x+4\] is positive for every \[x\in\mathbf{R}\]?
The function \[f(x)=x^3-3x^2+4x\], \[x\in\mathbf{R}\], is
For \[f(x)=4x^3-6x^2-72x+30\], which factorization of \[f'(x)\] is correct?
