Question

let f be a function such that f(2)=3 and f(2)= - 1. let g be the function g(x)=x². let g be a function defined as g(x)=f(x)·g(x). g(2)=□

Explanation:

Step1: Recall the product - rule

The product - rule states that if $G(x)=f(x)\cdot g(x)$, then $G^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)$.

Step2: Find $g^{\prime}(x)$

Given $g(x)=x^{2}$, by the power - rule $(x^{n})^\prime=nx^{n - 1}$, we have $g^{\prime}(x)=2x$.

Step3: Evaluate $g(2)$ and $g^{\prime}(2)$

Substitute $x = 2$ into $g(x)$ and $g^{\prime}(x)$. We get $g(2)=2^{2}=4$ and $g^{\prime}(2)=2\times2 = 4$.

Step4: Evaluate $f(2)$ and $f^{\prime}(2)$

Given $f(2)=3$ and $f^{\prime}(2)=-1$.

Step5: Evaluate $G^{\prime}(2)$

Substitute $x = 2$ into the product - rule formula $G^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)$.
$G^{\prime}(2)=f^{\prime}(2)g(2)+f(2)g^{\prime}(2)$.
$G^{\prime}(2)=(-1)\times4+3\times4$.
$G^{\prime}(2)=-4 + 12$.
$G^{\prime}(2)=8$.

Answer:

$8$