Advertisements
Advertisements
Question
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Advertisements
Solution
We know, y = [x (x – 2)]² = x² (x + 4 – 4x)
= x4 – 4x3 + 4x2
On differentiating with respect to x,
`dy/dx` = 4x2 - 12x2 + 8x
= 4x (x2 - 3x + 2)
= 4x ( x - 1) (x - 2)
`dy/dx` = 0
`=>` 4x ( x - 1) (x - 2) = 0
`therefore` x = 0, 1, 2
∴ Four parts of real number line from x = 0, x = 1, x = 2 are intervals.
(`- infty`, 0), (0, 1), (1, 2), (2, 2) are formed.
| Interval | (∞, 0) | (0, 1) | (1, 2) | (2, ∞) |
| Sign of x | -ve | +ve | +ve | +ve |
| sign of (x - 1) | -ve | - ve | +ve | +ve |
| sign of (x - 2) | -ve | - ve | -ve | +ve |
| sign of `dy/dx` | -ve | +ve | -ve | +ve |
| nature of function | decreasing | increasing | decreasing | increasing |
∴ y is an increasing function in (0, 1) ∪ (2, ∞)
APPEARS IN
RELATED QUESTIONS
Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R
Find the intervals in which the function f given by f(x) = 2x2 − 3x is
- strictly increasing
- strictly decreasing
Prove that y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`
Prove that the logarithmic function is strictly increasing on (0, ∞).
Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.
The interval in which y = x2 e–x is increasing is ______.
Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) decreasing.
Show that the function f(x) = 4x3 - 18x2 + 27x - 7 is always increasing on R.
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 15x2 + 36x + 1 ?
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) = −2x3 − 9x2 − 12x + 1 ?
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 12x2 + 36x + 17 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x3 + 4x2 + 15 ?
Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0 ?
Show that the function x2 − x + 1 is neither increasing nor decreasing on (0, 1) ?
What are the values of 'a' for which f(x) = ax is increasing on R ?
State whether f(x) = tan x − x is increasing or decreasing its domain ?
If the function f(x) = 2x2 − kx + 5 is increasing on [1, 2], then k lies in the interval
Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy.
Function f(x) = cos x − 2 λ x is monotonic decreasing when
If x = cos2 θ and y = cot θ then find `dy/dx at θ=pi/4`
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.
Find the values of x for which the following functions are strictly increasing:
f(x) = 3 + 3x – 3x2 + x3
Find the values of x for which the following func- tions are strictly increasing : f(x) = x3 – 6x2 – 36x + 7
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Show that function f(x) =`3/"x" + 10`, x ≠ 0 is decreasing.
Prove that function f(x) = `x - 1/x`, x ∈ R and x ≠ 0 is increasing function
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
A circular pIate is contracting at the uniform rate of 5cm/sec. The rate at which the perimeter is decreasing when the radius of the circle is 10 cm Jong is
A ladder 20 ft Jong leans against a vertical wall. The top-end slides downwards at the rate of 2 ft per second. The rate at which the lower end moves on a horizontal floor when it is 12 ft from the wall is ______
Given P(x) = x4 + ax3 + bx2 + cx + d such that x = 0 is the only real root of P'(x) = 0. If P(-1) < P(1), then in the interval [-1, 1] ______
The values of k for which the function f(x) = kx3 – 6x2 + 12x + 11 may be increasing on R are ______.
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) = tan-1 (sin x + cos x) is an increasing function in:
Find the interval in which the function `f` is given by `f(x) = 2x^2 - 3x` is strictly decreasing.
If f(x) = `x - 1/x`, x∈R, x ≠ 0 then f(x) is increasing.
If f(x) = x5 – 20x3 + 240x, then f(x) satisfies ______.
y = log x satisfies for x > 1, the inequality ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
