Find the inflection points, if any, of the function.
D) none

Question

Find the inflection points, if any, of the function. ​  ​
D) none

Quizplus · Accepted Answer

To find the inflection points of the function, we need to find the second derivative and set it equal to zero. Taking the derivative of the given function, we get:

f'(x) = 2x^3 - 6x^2 + 6x
f''(x) = 6x^2 - 12x + 6

Setting f''(x) = 0, we get:

6x^2 - 12x + 6 = 0
x^2 - 2x + 1 = 0
(x - 1)^2 = 0

So, the only critical point of the second derivative is x = 1. To determine whether x = 1 is an inflection point or not, we need to look at the sign of f''(x) on either side of x = 1. Taking a value of x slightly less than 1, say x = 0.9, we have:

f''(0.9) = 6(0.9)^2 - 12(0.9) + 6 = 1.56

Taking a value of x slightly greater than 1, say x = 1.1, we have:

f''(1.1) = 6(1.1)^2 - 12(1.1) + 6 = 0.36

Since f''(0.9) > 0 and f''(1.1) < 0, we can conclude that x = 1 is an inflection point. Therefore, the answer is choice C.

Find the Inflection Points, If Any, of the Function