SOLUTION: What is the maximum y value of V(x)=x(x-10)(x-8)

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Question 697551: What is the maximum y value of V(x)=x(x-10)(x-8)
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
What is the maximum y value of V(x)=x(x-10)(x-8)
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V(x) = x^3 - 18x^2 + 80x
V'(x) = 3x^2 - 36x + 80 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B-36x%2B80+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-36%29%5E2-4%2A3%2A80=336.

Discriminant d=336 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--36%2B-sqrt%28+336+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-36%29%2Bsqrt%28+336+%29%29%2F2%5C3+=+9.05505046330389
x%5B2%5D+=+%28-%28-36%29-sqrt%28+336+%29%29%2F2%5C3+=+2.94494953669611

Quadratic expression 3x%5E2%2B-36x%2B80 can be factored:
3x%5E2%2B-36x%2B80+=+%28x-9.05505046330389%29%2A%28x-2.94494953669611%29
Again, the answer is: 9.05505046330389, 2.94494953669611. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-36%2Ax%2B80+%29

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V''(x) = 6x - 36
--> inflection at x = 6
--> local max @ x2, 2.9449...
max = V(x2) =~ 105.0276