Do the functions have the same concavity?

Question

\[f(x)=-x^2-15x\]
A coordinate plane. The x- and y-axes both scale by one. The graph is the function y equals g of x which is a parabola that opens up. The function decreases through three, one and four, negative three. It has a minimum around five point five, negative five, then the function increases through seven, negative three and eight, one.
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{8}\]
\[\small{9}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{1}\]
\[\small{2}\]
\[\small{3}\]
\[\small{4}\]
\[\small{5}\]
\[\small{6}\]
\[\small{7}\]
\[\small{\llap{-}2}\]
\[\small{\llap{-}3}\]
\[\small{\llap{-}4}\]
\[\small{\llap{-}5}\]
\[\small{\llap{-}6}\]
\[\small{\llap{-}7}\]
\[y\]
\[x\]
\[y=g(x)\]
Choose 1 answer:
Choose 1 answer:
(Choice A) Yes,
\[f\] and
\[g\] are both concave down.
A
Yes,
\[f\] and
\[g\] are both concave down.
(Choice B) Yes,
\[f\] and
\[g\] are both concave up.
B
Yes,
\[f\] and
\[g\] are both concave up.
(Choice C) No,
\[f\] is concave up and
\[g\] is concave down.
C
No,
\[f\] is concave up and
\[g\] is concave down.
(Choice D) No,
\[f\] is concave down and
\[g\] is concave up.
D
No,
\[f\] is concave down and
\[g\] is concave up.

Answer 1

To determine the concavity of the functions \( f(x) \) and \( g(x) \), we need to analyze their second derivatives.

1. Function \( f(x) = -x^2 - 15x \):

   • First, we find the first derivative: \[ f'(x) = -2x - 15 \]
   • Next, we find the second derivative: \[ f''(x) = -2 \]
   • Since \( f''(x) = -2 \) is negative for all values of \( x \), the function \( f(x) \) is concave down.

2. Function \( g(x) \):

   • From the description in the problem, it states that the graph of \( g(x) \) opens upward, has a minimum point, and then increases. This indicates that \( g(x) \) is a parabola that is concave up.
   • Therefore, the concavity of \( g(x) \) is upward.

Now we can compare the concavity of \( f(x) \) and \( g(x) \):

• \( f(x) \) is concave down.
• \( g(x) \) is concave up.

From this analysis, we can conclude that the functions do not have the same concavity.

The correct answer is Choice C: No, \( f \) is concave down and \( g \) is concave up.