Question

let $f(x)=\frac{x^{2}+7x + 10}{2x + 4}$.
(a) $f(3)=$
(b) $f(3)=$
note: there are two ways to do this problem. the first is t and does not use the quotient rule.
question help: video

Explanation:

Step1: Simplify the function

First, factor the numerator $x^{2}+7x + 10=(x + 2)(x+5)$. Then $f(x)=\frac{(x + 2)(x + 5)}{2(x + 2)}=\frac{x + 5}{2}=\frac{1}{2}x+\frac{5}{2}$ for $x
eq - 2$.

Step2: Find the first - derivative

The derivative of $y = f(x)=\frac{1}{2}x+\frac{5}{2}$ using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$ (here $n = 1$ for the $x$ term) is $f'(x)=\frac{1}{2}$.

Step3: Evaluate the first - derivative at $x = 3$

Since $f'(x)$ is a constant function, $f'(3)=\frac{1}{2}$.

Step4: Find the second - derivative

The derivative of a constant function $y=f'(x)=\frac{1}{2}$ (where $n = 0$) is $f''(x)=0$.

Step5: Evaluate the second - derivative at $x = 3$

Since $f''(x)$ is a constant function, $f''(3)=0$.

Answer:

(a) $\frac{1}{2}$
(b) $0$