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# Increasing and decreasing functions watch

1. Q) Prove that the function f(x)= 2 -5x +2x^2 - x^3 is a decreasing function

So I know how to show this for a certain interval, but I don't understand how to prove it for the whole function?
This is A(S) level Maths by the way.
Thanks! Any and all suggestions or explanations helpful
Q) Prove that the function f(x)= 2 -5x +2x^2 - x^3 is a decreasing function

So I know how to show this for a certain interval, but I don't understand how to prove it for the whole function?
This is A(S) level Maths by the way.
Thanks! Any and all suggestions or explanations helpful
You need to show that f'(x) is always negative - try completing the square.
3. f'(0) is clearly -ve, so if you can show it has no roots it must always be -ve.

[Often the slickest approach to show a quadratic is always +ve (or -ve) is to complete the square (e.g. if you had f'(x) = x^2 - 4x + 9, then f'(x) = (x-2)^2 + 5 so is always +ve). But it would be a little messy here].

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Updated: October 16, 2017
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