Increasing and decreasing functions: find the intervals where f(x)is increasing or decreasing. f(x)=x-x^2. I believe the answer is increasing (1/2,inf)
so far I have x-x^2
and I figured that meant the function was increasing at (1/2,infinity)
I'm not sure I'm understanding this right, so I want to know if what I've done is right, or how it's supposed to be done.
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`f(x) = x-x^2`
For any function maximum or minimum points can be found by the first derivative.
`(df(x))/dx = 1-2x`
At maximum and minimum points `(df(x))/dx = 0 `
`(df(x))/dx = 0`
`1-2x = 0`
` x = 1/2`
So at x = 1/2 there is a point of a graph which is a maximum of minimum.
If `(d^2f(x))/(dx^2) < 0` then the point is a maximum point and If `(d^2f(x))/(dx^2) > 0` then the point is a minimum point.
`(d^2f(x))/(dx^2) = -2 <0`
Therefore at x=1/2 we have a maximum point. So before x = 1/2 the function is increasing and after x = 1/2 the function is decreasing.
f(x) is increasing when `x in (-oo,1/2)`
f(x) is decreasing when `x in (1/2,-oo)`
For further clarification you can see the graph of `f(x) = x-x^2` below.
Thank you, that was one of the answers that I had gotten, It's good to see that I am doing something rightish =D
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