Question

question 3 (1 point)
suppose f and g are functions with the following properties:

$f(3)=g(3)=0,f(3)=f(3)=2,g(3)=g(3)=1.$

if the second - degree taylor polynomial of $y = f(x)g(x)$ about the point $x = 3$ is given by

$a_0 + a_1(x - 3)+a_2(x - 3)^2$

then $a_2$ is equal to which of the following?

0
2
4
-2
-1

Explanation:

Step1: Recall product - rule for derivatives

The first - derivative of $y = f(x)g(x)$ using the product rule $(uv)^\prime=u^\prime v + uv^\prime$ is $y^\prime=f^\prime(x)g(x)+f(x)g^\prime(x)$.
The second - derivative of $y = f(x)g(x)$ using the product rule on $y^\prime$:

$$
LATEXBLOCK0
$$

Step2: Evaluate $y^{\prime\prime}$ at $x = 3$

We know that $f(3)=g(3)=0$, $f^\prime(3)=f^{\prime\prime}(3)=2$, and $g^\prime(3)=g^{\prime\prime}(3)=1$.
Substitute $x = 3$ into the formula for $y^{\prime\prime}$:

$$
LATEXBLOCK1
$$

Step3: Recall the formula for the Taylor polynomial

The second - degree Taylor polynomial of a function $y = h(x)$ about $x = a$ is $P_2(x)=h(a)+h^\prime(a)(x - a)+\frac{h^{\prime\prime}(a)}{2}(x - a)^2$.
For $y = f(x)g(x)$ about $x = 3$, $a_2=\frac{y^{\prime\prime}(3)}{2}$.
Since $y^{\prime\prime}(3)=4$, then $a_2 = 2$.

Answer:

2