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
The interval in which y = x2 e–x is increasing is ______.
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.
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 9x2 + 12x − 5 ?
Find the interval in which the following function are increasing or decreasing f(x) = x8 + 6x2 ?
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) ?
Show that f(x) = tan−1 x − x is a decreasing function on R ?
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 given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
Show that f(x) = x + cos x − a is an increasing function on R for all values of a ?
Write the set of values of a for which the function f(x) = ax + b is decreasing for all x ∈ R ?
If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then
Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).
Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.
For manufacturing x units, labour cost is 150 – 54x and processing cost is x2. Price of each unit is p = 10800 – 4x2. Find the value of x for which Total cost is decreasing.
The total cost of manufacturing x articles is C = 47x + 300x2 − x4. Find x, for which average cost is increasing.
Find the intervals in which the function `f("x") = (4sin"x")/(2+cos"x") -"x";0≤"x"≤2pi` is strictly increasing or strictly decreasing.
The edge of a cube is decreasing at the rate of`( 0.6"cm")/sec`. Find the rate at which its volume is decreasing, when the edge of the cube is 2 cm.
Test whether the following functions are increasing or decreasing : f(x) = 2 – 3x + 3x2 – x3, x ∈ R.
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
A man of height 1.9 m walks directly away from a lamp of height 4.75m on a level road at 6m/s. The rate at which the length of his shadow is increasing is
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
2x3 - 6x + 5 is an increasing function, if ____________.
The function f(x) = tan-1 (sin x + cos x) is an increasing function in:
The function `"f"("x") = "log" (1 + "x") - (2"x")/(2 + "x")` is increasing on ____________.
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 f(x) be a function such that; f'(x) = log1/3(log3(sinx + a)) (where a ∈ R). If f(x) is decreasing for all real values of x then the exhaustive solution set of a is ______.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
