SOLUTION: In a cubic graph, how do I find the coordinates of the turning points. Eg, xcubed - 3xsquared - 4x + 12. Can I use differentiation?

Algebra ->  Graphs -> SOLUTION: In a cubic graph, how do I find the coordinates of the turning points. Eg, xcubed - 3xsquared - 4x + 12. Can I use differentiation?      Log On


   

Question 172220: In a cubic graph, how do I find the coordinates of the turning points.
Eg, xcubed - 3xsquared - 4x + 12. Can I use differentiation?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
In a cubic graph, how do I find the coordinates of the turning points.
Eg, xcubed - 3xsquared - 4x + 12. Can I use differentiation?
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Setting the 1st derivative = 0 will give the max and mins (if any) of the cubic.
d/dx of x^3 - 3x^2 - 4x + 12 = 3x^2 - 6x - 4
3x^2 - 6x - 4 = 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-6x%2B-4+=+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=%28-6%29%5E2-4%2A3%2A-4=84.

Discriminant d=84 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--6%2B-sqrt%28+84+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+84+%29%29%2F2%5C3+=+2.52752523165195
x%5B2%5D+=+%28-%28-6%29-sqrt%28+84+%29%29%2F2%5C3+=+-0.527525231651947

Quadratic expression 3x%5E2%2B-6x%2B-4 can be factored:
3x%5E2%2B-6x%2B-4+=+%28x-2.52752523165195%29%2A%28x--0.527525231651947%29
Again, the answer is: 2.52752523165195, -0.527525231651947. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-6%2Ax%2B-4+%29

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The 1st derivative has 2 zeros. That means there are 2 points of inflection, a local max and a local min.