Advertisements
Advertisements
प्रश्न
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
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 intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
Test whether the function is increasing or decreasing.
f(x) = `"x" -1/"x"`, x ∈ R, x ≠ 0,
Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`
Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.
Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R ?
Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?
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) = 2x3 − 24x + 7 ?
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) ?
Prove that the following function is increasing on R f \[(x) =\]3 \[x^5\] + 40 \[x^3\] + 240\[x\] ?
Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π).
What are the values of 'a' for which f(x) = ax is decreasing on R ?
The function \[f\left( x \right) = \log_e \left( x^3 + \sqrt{x^6 + 1} \right)\] is of the following types:
The function \[f\left( x \right) = \frac{\lambda \sin x + 2 \cos x}{\sin x + \cos x}\] is increasing, if
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) = 2x3 - 15x2 + 36x + 1
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
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
If f(x) = x3 – 15x2 + 84x – 17, then ______.
Let f be a real valued function defined on (0, 1) ∪ (2, 4) such that f '(x) = 0 for every x, then ____________.
The function f(x) = x3 + 6x2 + (9 + 2k)x + 1 is strictly increasing for all x, if ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
State whether the following statement is true or false.
If f'(x) > 0 for all x ∈ (a, b) then f(x) is decreasing function in the interval (a, b).
y = log x satisfies for x > 1, the inequality ______.
The intevral in which the function f(x) = 5 + 36x – 3x2 increases will be ______.
Find the interval in which the function f(x) = x2e–x is strictly increasing or decreasing.
