Question

How do you find the intervals where the function `f(x)=(2x-1^2)(x-3)^2` is concave up and concave down?

Answer 1

Concave down in `(-oo,13/6)` and concave up in `(13/6,oo)`

For finding the intervals where f(x) is concave up or concave down, first get f'' (x).

f(x) is `(2x-1)(x-3)^2 = 2x^3 - 13x^2 +24x -9`

f'(x)=`6x^2-26x +24` and f'' (x)= 12x-26. The inflection point is at x=`13/6`

For `x<=13/6` , f''(x) is negative, hence f(x) would be concave down in the interval `(-oo, 13/6)`

For `x>= 13/6`, f''(x) is positive, hence f(x) would be concave up in the interval `(13/6, oo)`