Advertisements
Advertisements
प्रश्न
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Advertisements
उत्तर १
`f(x) = x^3 - 3x^2 + 6x - 100`
`f'(x) = 3x^2 - 6x + 6`
`= 3(x^2 - 2x + 1 ) + 3`
=`3(x+1)^2 + 3 > 0`
For all values of x, `(x - 1)^2` is always positve
`:. f'(x) > 0`
So, f (x) is increasing function.
उत्तर २
The given function is
f(x) = x3 − 3x2 + 6x −100
∴f'(x) = 3x2 − 6x + 6
=3(x2 − 2x +2)
=3(x2 − 2x + 1) + 3
=3(x−1)2+3
For f(x) to be increasing, we must have f'(x) > 0
Now, 3(x−1)2 ≥ 0 ∀x ∈ R
⇒ 3(x − 1)2 + 3 > 0 ∀x ∈ R
⇒ f'(x) > 0 ∀x ∈ R
Hence, the given function is increasing on R
संबंधित प्रश्न
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Find the intervals in which the following functions are strictly increasing or decreasing:
10 − 6x − 2x2
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
Water is dripping out from a conical funnel of semi-verticle angle `pi/4` at the uniform rate of `2 cm^2/sec`in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.
Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 107 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
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) = sin x − cos x is an increasing function on (−π/4, π/4)?
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) ?
Write the set of values of 'a' for which f(x) = loga x is increasing in its domain ?
The function f(x) = cot−1 x + x increases in the interval
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
The function \[f\left( x \right) = \frac{x}{1 + \left| x \right|}\] is
Function f(x) = loga x is increasing on R, if
Show that the function f given by f(x) = tan–1 (sin x + cos x) is decreasing for all \[x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) .\]
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
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 decreasing function.
f(x) = 2x3 - 15x2 - 144x - 7
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
The sides of a square are increasing at the rate of 0.2 cm/sec. When the side is 25cm long, its area is increasing at the rate of ______
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
Which of the following graph represent the strictly increasing function.
y = log x satisfies for x > 1, the inequality ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
If \[f'(x)<0\] throughout an interval, then \[f\] is
As one moves from left to right on a graph, what do constant \[y\]-values indicate?
