do the functions have the same concavity?\n$f(x)=x^{2}-16x + 48$\nchoose 1 answer:\na yes, $f$ and $g$ are…

do the functions have the same concavity?\n$f(x)=x^{2}-16x + 48$\nchoose 1 answer:\na yes, $f$ and $g$ are both concave down.\nb yes, $f$ and $g$ are both concave up.\nc no, $f$ is concave up and $g$ is concave down.\nd no, $f$ is concave down and $g$ is concave up.

do the functions have the same concavity?\n$f(x)=x^{2}-16x + 48$\nchoose 1 answer:\na yes, $f$ and $g$ are both concave down.\nb yes, $f$ and $g$ are both concave up.\nc no, $f$ is concave up and $g$ is concave down.\nd no, $f$ is concave down and $g$ is concave up.

Answer

Explanation:

Step1: Find the second - derivative of $f(x)$.

First, find the first - derivative of $f(x)=x^{2}-16x + 48$. Using the power rule $(x^n)^\prime=nx^{n - 1}$, we have $f^\prime(x)=2x-16$. Then, find the second - derivative $f^{\prime\prime}(x)=(2x - 16)^\prime=2$.

Step2: Analyze the concavity of $f(x)$.

Since $f^{\prime\prime}(x)=2>0$, the function $f(x)$ is concave up.

Step3: Analyze the concavity of $g(x)$ from the graph.

The graph of $y = g(x)$ is a parabola opening upwards. For a parabola opening upwards, the second - derivative is positive, so $g(x)$ is concave up.

Answer:

B. Yes, $f$ and $g$ are both concave up.