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For what values of x is #f(x)=(x-1)(x-7)(x-1)# concave or convex?

1 Answer
Sep 17, 2017

Answer:

Convex if x is greater than 1.5, concave if less than 1.5

Explanation:

A function is convex (aka concave up) when #f''(x)>0#, and concave aka concave down when <0. Thus, we must find the second derivative. We first multiply these factors...

#f (x)=(x-1)(x-7)(x-1) = (x^2-2x+1)(x-7) = x^3 - 7x^2 - 2x^2 +14x +x-7 = x^3 - 9x^2 + 15x -7 = f (x)#

Now we differentiate using the power rule...

#f'(x) =3x^2-9x+15#

And once more...

#f''(x) = 6x -9#

We then find when this is greater than 0...

#6x-9>0 -> 6x>9 -> x>1.5#

This, the function is convex for x greater than 1.5, concave for x less than 1.5