Question

question 2 - 1 point consider the function f(x) below. at what value(s) of x are the inflection points? enter answer using exact values, and separate multiple answers with a comma if necessary. f(x)=3x^5 - 4x^3 - 10x + 8 provide your answer below:

Explanation:

Step1: Find the first - derivative

Using the power rule $\frac{d}{dx}(ax^n)=nax^{n - 1}$, for $f(x)=3x^{5}-4x^{3}-10x + 8$, we have $f'(x)=15x^{4}-12x^{2}-10$.

Step2: Find the second - derivative

Differentiate $f'(x)$ again. $f''(x)=60x^{3}-24x$.

Step3: Set the second - derivative equal to zero

Set $f''(x) = 0$, so $60x^{3}-24x=0$. Factor out $12x$: $12x(5x^{2}-2)=0$.

Step4: Solve for $x$

From $12x(5x^{2}-2)=0$, we have two cases.
Case 1: $12x = 0$, then $x = 0$.
Case 2: $5x^{2}-2=0$, then $x^{2}=\frac{2}{5}$, and $x=\pm\frac{\sqrt{10}}{5}$.

Answer:

$0,\frac{\sqrt{10}}{5},-\frac{\sqrt{10}}{5}$