Question

do the functions have the same concavity?
$f(x)=-3x^{2}+12x - 32$
choose 1 answer:

Explanation:

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

For $f(x)=-3x^{2}+12x - 32$, first find the first - derivative $f^\prime(x)$ using the power rule. If $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. So $f^\prime(x)=-6x + 12$. Then find the second - derivative $f^{\prime\prime}(x)=-6$. Since $f^{\prime\prime}(x)<0$, the function $f(x)$ is concave down.

Step2: Analyze the concavity of $g(x)$ from the graph.

The graph of $y = g(x)$ is a parabola opening downwards. For a parabola opening downwards, the second - derivative is negative, and the function is concave down.

Answer:

Yes