calculate smallest and highest value of the function $\displaystyle y=x^3-2x*|x-2|$ on segment [0,3]

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- Jun 28th 2010, 11:09 AMGarascalculate smallest and highest value of the function
calculate smallest and highest value of the function $\displaystyle y=x^3-2x*|x-2|$ on segment [0,3]

- Jun 28th 2010, 10:24 PMecnanif
You check the values of the function at the endpoints, that is at the points $\displaystyle x=0 $ & $\displaystyle x=3 $.

Now, since your function is not differentiable at $\displaystyle x=2$, you cannot use the derivative at $\displaystyle x=2$.

So if you limit yourself to $\displaystyle (0,2)$ and $\displaystyle (2,3)$ when performing you differentiate and look for max/min, and check the function value at $\displaystyle x=2$.

Now you compare all the values you have. That is, you have your endpoint values, your value at $\displaystyle x=2$, and possibly some values at local max/min which you may have found during the differentiation. - Jun 28th 2010, 10:43 PMCaptainBlack
- Jun 29th 2010, 06:53 AMHallsofIvy
If x< 2, x-2< 0 so |x-2|= -(x-2)= 2- x. Then $\displaystyle y= x^3- 2x(2- x)= x^3+ 2x^2- 4x= x(x^2+ 2x- 4)$. You can complete the square to determine where $\displaystyle x^2+ 2x- 4$ has max or min.

If x> 2, x-2> 0 so |x-2|= x- 2. Then $\displaystyle y= x^3-2x(x-2)= x^3- 2x^2+ 4x= x(x^2- 2x+ 4)$. Again, try completing the square in $\displaystyle x^2- 2x+ 4$.