-/1 points details my notes tanapcalc10 4.2.052.mi. find the inflection point, if it exists, of the…

Explanation:

Step1: Find the first - derivative

Using the power rule $(x^n)'=nx^{n - 1}$, for $g(x)=4x^{3}-6x^{2}+6x - 5$, we have $g'(x)=12x^{2}-12x + 6$.

Step2: Find the second - derivative

Differentiate $g'(x)$ again. $g''(x)=24x-12$.

Step3: Set the second - derivative equal to zero

Set $g''(x) = 0$, so $24x-12 = 0$. Solving for $x$ gives $24x=12$, then $x=\frac{1}{2}$.

Step4: Find the $y$ - coordinate

Substitute $x = \frac{1}{2}$ into the original function $g(x)$. $g(\frac{1}{2})=4(\frac{1}{2})^{3}-6(\frac{1}{2})^{2}+6(\frac{1}{2})-5=4\times\frac{1}{8}-6\times\frac{1}{4}+3 - 5=\frac{1}{2}-\frac{3}{2}+3 - 5=-3$.

Answer:

$(\frac{1}{2},-3)$