Advertisements
Advertisements
प्रश्न
State whether the following statement is True or False:
The function f(x) = `"x"*"e"^("x" (1 - "x"))` is increasing on `((-1)/2, 1)`.
पर्याय
True
False
Advertisements
उत्तर
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
संबंधित प्रश्न
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Find the value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Prove that the function f given by f(x) = x2 − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).
Prove that the function f(x) = loge x is increasing on (0, ∞) ?
Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R ?
Find the interval in which the following function are increasing or decreasing f(x) = 6 + 12x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 24x + 7 ?
Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5 ?
Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0 ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R ?
What are the values of 'a' for which f(x) = ax is decreasing on R ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
The function f(x) = x2 e−x is monotonic increasing 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 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
Find the values of x for which f(x) = 2x3 – 15x2 – 144x – 7 is
- Strictly increasing
- strictly decreasing
The slope of tangent at any point (a, b) is also called as ______.
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
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 function f(x) = sin x + 2x is ______
The function which is neither decreasing nor increasing in `(pi/2,(3pi)/2)` is ____________.
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) = `x/sqrt(a^2 + x^2) - (d - x)/sqrt(b^2 + (d - x)^2), x ∈ R` where a, b and d are non-zero real constants. Then ______.
For \[f(x)=x^3-3x^2+4x\], what is \[f'(x)\]?
