Advertisements
Advertisements
Question
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
Options
True
False
Advertisements
Solution
True.
Explanation:
f(x) = `"x"*"e"^("x" (1 - "x"))`
∴ f '(x) = `"e"^("x" (1 - "x")) + "x"*"e"^("x" (1 - "x")) [1 - 2"x"]`
`= "e"^("x" (1 - "x")) [1 + "x" - 2"x"^2]`
If f(x) is increasing, then f '(x) > 0.
Consider f '(x) > 0
∴ `"e"^("x" (1 - "x")) (1 + "x" - 2"x"^2)` > 0
∴ 2x2 - x - 1 < 0
∴ (2x + 1)(x - 1) < 0
ab < 0 ⇔ a > 0 and b < 0 or a < 0 or b > 0
∴ Either (2x + 1) > 0 and (x – 1) < 0 or
(2x + 1) < 0 and (x – 1) > 0
Case 1: (2x + 1) > 0 and (x – 1) < 0
∴ x > `-1/2` and x < 1
i.e., x ∈ `(-1/2, 1)`
Case 2: (2x + 1) < 0 and (x – 1) > 0
∴ x < `- 1/2` and x > 1
which is not possible.
∴ f(x) is increasing on `(-1/2, 1)`
APPEARS IN
RELATED QUESTIONS
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
The function f (x) = x3 – 3x2 + 3x – 100, x∈ R is _______.
(A) increasing
(B) decreasing
(C) increasing and decreasing
(D) neither increasing nor decreasing
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)`
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(x) = 5 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?
Find the interval in which f(x) is increasing or decreasing f(x) = x|x|, x \[\in\] R ?
Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < \[\frac{\pi}{2}\] ?
Find 'a' for which f(x) = a (x + sin x) + a is increasing on R ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Every invertible function is
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
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
Find the values of x for which the function f(x) = x3 – 12x2 – 144x + 13 (a) increasing (b) decreasing
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
If the function f(x) = `7/x - 3`, x ∈ R, x ≠ 0 is a decreasing function, then x ∈ ______
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R
The function f(x) = 4 sin3x – 6 sin2x + 12 sinx + 100 is strictly ______.
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f (x) = x2, for all real x, is ____________.
Which of the following graph represent the strictly increasing function.
Find the value of x for which the function f(x)= 2x3 – 9x2 + 12x + 2 is decreasing.
Given f(x) = 2x3 – 9x2 + 12x + 2
∴ f'(x) = `squarex^2 - square + square`
∴ f'(x) = `6(x - 1)(square)`
Now f'(x) < 0
∴ 6(x – 1)(x – 2) < 0
Since ab < 0 ⇔a < 0 and b < 0 or a > 0 and b < 0
Case 1: (x – 1) < 0 and (x – 2) < 0
∴ x < `square` and x > `square`
Which is contradiction
Case 2: x – 1 and x – 2 < 0
∴ x > `square` and x < `square`
1 < `square` < 2
f(x) is decreasing if and only if x ∈ `square`
Let 'a' be a real number such that the function f(x) = ax2 + 6x – 15, x ∈ R is increasing in `(-∞, 3/4)` and decreasing in `(3/4, ∞)`. Then the function g(x) = ax2 – 6x + 15, x∈R has a ______.
If f(x) = x3 + 4x2 + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
If \[f'(x)>0\] throughout an interval, then \[f\] is
As one moves from left to right on a graph, what does an increase in the \[y\]-values indicate?
In the procedure for finding intervals, which conclusion is correct when \[f'(x)<0\]?
