Question

let f be a differentiable function such that f(2)=2 and f(2)=5. if g(x)=x³f(x), what is the value of g(2)?

Explanation:

Step1: Apply product - rule

The product - rule states that if $g(x)=x^{3}f(x)$, then $g^{\prime}(x)=(x^{3})^{\prime}f(x)+x^{3}f^{\prime}(x)$.
Since $(x^{3})^{\prime}=3x^{2}$, we have $g^{\prime}(x)=3x^{2}f(x)+x^{3}f^{\prime}(x)$.

Step2: Substitute $x = 2$

We know that $f(2) = 2$ and $f^{\prime}(2)=5$.
Substitute $x = 2$ into $g^{\prime}(x)$:
$g^{\prime}(2)=3\times2^{2}\times f(2)+2^{3}\times f^{\prime}(2)$.

Step3: Calculate the value

First, calculate $3\times2^{2}\times f(2)$:
$3\times2^{2}\times f(2)=3\times4\times2 = 24$.
Second, calculate $2^{3}\times f^{\prime}(2)$:
$2^{3}\times f^{\prime}(2)=8\times5 = 40$.
Then $g^{\prime}(2)=24 + 40=64$.

Answer:

$64$