sketch a graph of a function f with the following properties. f<0 and f<0, for x<22 f<0 and f>0, for x>22…

Explanation:

Step1: Analyze $x < 22$

Since $f'<0$ and $f''<0$ for $x < 22$, the function is decreasing and concave - down. A decreasing function has a negative slope and a concave - down function curves downwards.

Step2: Analyze $x>22$

Since $f'<0$ and $f''>0$ for $x > 22$, the function is decreasing and concave - up. A decreasing function has a negative slope and a concave - up function curves upwards.

Step3: Check the graphs

• Option A: The function is increasing for $x < 22$ which does not match the condition $f'<0$ for $x < 22$, so it is incorrect.
• Option B: The function is increasing for $x < 22$ which does not match the condition $f'<0$ for $x < 22$, so it is incorrect.
• Option C: For $x < 22$, the function is decreasing and concave - down, and for $x>22$ the function is decreasing and concave - up, which matches the given conditions.
• Option D: The function is increasing for $x > 22$ which does not match the condition $f'<0$ for $x > 22$, so it is incorrect.

Answer:

C.