Question

find the first and second derivatives of the function f(x)=x^4 - 24x^2 + 64x + 16 and report the x - coordinates of any relative extrema and inflection points.
the graphs of f(x), f(x), and f(x) are shown to the right.
f(x)=4x^3 - 48x + 64
f(x)=12x^2 - 48
b. complete
the function f(x) has a relative minimum at x = - 4
the inflection points are at x = (use a comma to separate answers as needed)

Explanation:

Step1: Recall inflection - point condition

Inflection points occur where $f''(x)=0$.

Step2: Set $f''(x)$ equal to 0

Given $f''(x)=12x^{2}-48$, set $12x^{2}-48 = 0$.

Step3: Solve the equation

First, factor out 12: $12(x^{2}-4)=0$. Then, since $x^{2}-4=(x - 2)(x + 2)$, we have $(x - 2)(x + 2)=0$. Solving for $x$, we get $x=2$ or $x=-2$.

Answer:

$2,-2$