SOLUTION: In the function f(x) = ax 2 + bx + c, the minimum or maximum value occurs where x is equal to -b/(2a). How is that value derived?

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Question 384359: In the function f(x) = ax 2 + bx + c, the minimum or maximum value occurs where x is equal to -b/(2a). How is that value derived?

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Good question. However, the proof requires a little introductory calculus.
In calculus, we define a function called the "derivative," which measures the instantaneous rate of change of a function. Basically, it is the same as finding slope, except that the two points get infinitely close, and it can be evaluated using limits. The derivative is usually denoted f'(x) or %28d%2Fdx%29+f%28x%29.
By the power rule (you'll learn it early in calculus), the derivative of ax%5E2+%2B+bx+%2B+c is 2ax+%2B+b. The relative minimum or maximum occurs when the derivative is equal to zero, and the slope is positive on one side of that point and negative on the other.
Since the derivative of the quadratic is 2ax+%2B+b it is easy to see that this derivative is equal to zero when x+=+-b%2F2a, therefore it is the vertex.