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
Find the intervals in which the following functions are strictly increasing or decreasing:
−2x3 − 9x2 − 12x + 1
Find the intervals in which the following functions are strictly increasing or decreasing:
(x + 1)3 (x − 3)3
Show that y = `log(1+x) - (2x)/(2+x), x> - 1`, is an increasing function of x throughout its domain.
Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).
Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function 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) = 8 + 36x + 3x2 − 2x3 ?
Find the interval in which the following function are increasing or decreasing f(x) = x4 − 4x ?
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{x^4}{4} + \frac{2}{3} x^3 - \frac{5}{2} x^2 - 6x + 7\] ?
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 an increasing function on (−π/2, π/2) ?
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
Write the set of values of k for which f(x) = kx − sin x is increasing on R ?
The interval of increase of the function f(x) = x − ex + tan (2π/7) is
The function f(x) = xx decreases on the interval
The consumption expenditure Ec of a person with the income x. is given by Ec = 0.0006x2 + 0.003x. Find MPC, MPS, APC and APS when the income x = 200.
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.
Show that function f(x) =`("x - 2")/("x + 1")`, x ≠ -1 is increasing.
Find the values of x for which the function f(x) = 2x3 – 6x2 + 6x + 24 is strictly increasing
State whether the following statement is True or False:
The function f(x) = `3/x` + 10, x ≠ 0 is decreasing
Show that the function f(x) = `(x - 2)/(x + 1)`, x ≠ – 1 is increasing
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
The function f(x) = x2 – 2x is increasing in the interval ____________.
Let f (x) = tan x – 4x, then in the interval `[- pi/3, pi/3], "f"("x")` is ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
Let x0 be a point in the domain of definition of a real valued function `f` and there exists an open interval I = (x0 – h, ro + h) containing x0. Then which of the following statement is/ are true for the above statement.
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 ______.
Let f: [0, 2]→R be a twice differentiable function such that f"(x) > 0, for all x ∈( 0, 2). If `phi` (x) = f(x) + f(2 – x), then `phi` is ______.
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 ______.
Function f(x) = x100 + sinx – 1 is increasing for all x ∈ ______.
The interval in which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Let f(x) = x3 – 6x2 + 9x + 18, then f(x) is strictly increasing in ______.
The function f(x) = sin4x + cos4x is an increasing function if ______.
Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.
If \[f'(x)<0\] throughout an interval, then \[f\] is
As one moves from left to right on a graph, what does a decrease in the \[y\]-values indicate?
Which form shows that \[f'(x)=3x^2-6x+4\] is positive for every \[x\in\mathbf{R}\]?
