Question

Consider the function y = 3x⁴ – 24x³ + 72x². Which of the following statements is true?

A - There are inflection points at x = 2 and x = -2
B - There are inflection points at x = 1 and x = -1
C - There is an inflection point only at x = 1
D - There are no inflection points

Answer 1

After computing the second derivative of the function y = 3x⁴ – 24x³ + 72x², it is determined that there are no inflection points because the second derivative does not change sign.

Step-by-step explanation:

To determine the inflection points of the function y = 3x⁴ – 24x³ + 72x², we need to find the second derivative and analyze where it changes sign.

First, compute the first derivative:
y' = 12x³ – 72x² + 144x.

Then, find the second derivative:
y'' = 36x² – 144x + 144.

To find potential inflection points, set y'' equal to zero:
0 = 36x² – 144x + 144. This simplifies to x² – 4x + 4 = 0, which factors to (x – 2)² = 0. Therefore, we have a potential inflection point at x = 2.

However, since (x – 2)² is always nonnegative for real numbers, y'' does not change sign around x = 2, signifying that there is no actual inflection point at x = 2. As this is the only potential inflection point we get from setting the second derivative to zero, we conclude there are no inflection points for this function.