Advertisements
Advertisements
Question
The function f(x) = x2 e−x is monotonic increasing when
Options
x ∈ R − [0, 2]
0 < x < 2
2 < x < ∞
x < 0
Advertisements
Solution
0 < x < 2
\[f\left( x \right) = x^2 e^{- x} \]
\[f'\left( x \right) = 2x e^{- x} - x^2 e^{- x} \]
\[ = e^{- x} x\left( 2 - x \right)\]
\[\text { For f(x) to be monotonic increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow e^{- x} x\left( 2 - x \right) > 0 \left[ \because e^{- x} > 0 \right]\]
\[ \Rightarrow x\left( 2 - x \right) > 0\]
\[ \Rightarrow x\left( x - 2 \right) < 0\]
\[ \Rightarrow 0 < x < 2\]
APPEARS IN
RELATED QUESTIONS
Find the intervals in which the following functions are strictly increasing or decreasing:
x2 + 2x − 5
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Which of the following functions are strictly decreasing on `(0, pi/2)`?
- cos x
- cos 2x
- cos 3x
- tan x
Find the least value of a such that the function f given by f (x) = x2 + ax + 1 is strictly increasing on [1, 2].
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing 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) = 2x3 − 12x2 + 18x + 15 ?
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) = \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\] x > 0 ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \log\left( 2 + x \right) - \frac{2x}{2 + x}, x \in R\] ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
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) ?
Find the intervals in which f(x) = (x + 2) e−x is increasing or decreasing ?
Prove that the following function is increasing on R f \[f\left( x \right) = 4 x^3 - 18 x^2 + 27x - 27\] ?
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 increasing on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
Function f(x) = cos x − 2 λ x is monotonic decreasing when
f(x) = 2x − tan−1 x − log \[\left\{ x + \sqrt{x^2 + 1} \right\}\] is monotonically increasing when
Find the intervals in which the function \[f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] is
(a) strictly increasing
(b) strictly decreasing
Find the values of x for which the following functions are strictly decreasing:
f(x) = 2x3 – 3x2 – 12x + 6
Find the values of x for which the following functions are strictly decreasing : f(x) = `x + (25)/x`
Test whether the function f(x) = x3 + 6x2 + 12x − 5 is increasing or decreasing for all x ∈ R
Find the values of x, for which the function f(x) = x3 + 12x2 + 36𝑥 + 6 is monotonically decreasing
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
For every value of x, the function f(x) = `1/"a"^x`, a > 0 is ______.
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 ______
For which interval the given function f(x) = 2x3 – 9x2 + 12x + 7 is increasing?
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The function `"f"("x") = "x"/"logx"` increases on the interval
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
If f(x) = x + cosx – a then ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
