Advertisements
Advertisements
Question
Find the value(s) of x for which y = [x(x − 2)]2 is an increasing function.
Advertisements
Solution
f(x)=[x(x−2)]2
f'(x)=2[x(x−2)]{x−2+x}
f'(x)=4x(x−2)(x−1)
At critical point, f'(x)=0
4x(x−2)(x−1)=0
⇒x=0,1,2
| Interval | f'(x)=4x(x−1)(x−2) |
Result |
| (−∞,0) | f'(−1)=4(−1)(−2)(−3)=−24<0 | Decreasing |
| (0,1) | f'(1/2)=4(1/2)(−1/2)(−3/2)=3/2>0 | Increasing |
| (1,2) | f'(3/2)=4(3/2)(1/2)(−1/2)=−3/2<0 | Decreasing |
| (2,∞) | f'(3)=4(3)(2)(1)=24>0 | Increasing |
So, the function is increasing in the interval (0,1)∪(2,∞).
APPEARS IN
RELATED QUESTIONS
Show that the function given by f(x) = sin x is
- strictly increasing in `(0, pi/2)`
- strictly decreasing in `(pi/2, pi)`
- neither increasing nor decreasing in (0, π)
Find the values of x for `y = [x(x - 2)]^2` is an increasing function.
Find the interval in which the following function are increasing or decreasing f(x) = 5x3 − 15x2 − 120x + 3 ?
Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing ?
Show that the function f(x) = cot \[-\] l(sinx + cosx) is decreasing on \[\left( 0, \frac{\pi}{4} \right)\] and increasing on \[\left( 0, \frac{\pi}{4} \right)\] ?
The function f(x) = xx decreases on the interval
Function f(x) = loga x is increasing on R, if
Let ϕ(x) = f(x) + f(2a − x) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
If the function f(x) = x2 − kx + 5 is increasing on [2, 4], then
The function f(x) = x9 + 3x7 + 64 is increasing on
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).
If the demand function is D = 50 - 3p - p2, find the elasticity of demand at (a) p = 5 (b) p = 2 , Interpret your result.
Find the values of x for which the following functions are strictly increasing : f(x) = 2x3 – 3x2 – 12x + 6
Show that y = `log (1 + x) – (2x)/(2 + x), x > - 1` is an increasing function on its domain.
The slope of tangent at any point (a, b) is also called as ______.
Find the values of x such that f(x) = 2x3 – 15x2 + 36x + 1 is increasing 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
If f(x) = [x], where [x] is the greatest integer not greater than x, then f'(1') = ______.
In which interval is the given function, f(x) = 2x3 - 21x2 + 72x + 19 monotonically decreasing?
For every value of x, the function f(x) = `1/7^x` is ______
Show that f(x) = 2x + cot–1x + `log(sqrt(1 + x^2) - x)` is increasing in R
The function f(x) = mx + c where m, c are constants, is a strict decreasing function for all `"x" in "R"` , if ____________.
The interval in which the function f is given by f(x) = x2 e-x is strictly increasing, is: ____________.
`"f"("x") = (("e"^(2"x") - 1)/("e"^(2"x") + 1))` is ____________.
The function `"f"("x") = "x"/"logx"` increases on the interval
The length of the longest interval, in which the function `3 "sin x" - 4 "sin"^3"x"` is increasing, is ____________.
Function f(x) = `log(1 + x) - (2x)/(2 + x)` is monotonically increasing when ______.
In which one of the following intervals is the function f(x) = x3 – 12x increasing?
