Provide an appropriate response.
-$$\text { Determine the values of constants a and b so that } f ( x ) = a x ^ { 2 } + b x \text { has an absolute maximum at the point } ( 2,4 ) \text {. }$$
A) a = 1, b = 4
B) a = 1, b = 2
C) a = -1, b = 2
D) a = -1, b = 4

Question

Quizplus · Accepted Answer

To find the values of a and b, we need to use the fact that the absolute maximum occurs at (2,4). This means that both the first and second derivatives of f(x) should be equal to zero at x=2.

The first derivative of f(x) is: 
f'(x) = 2ax + b

Substituting x=2, we get: 
4a + b = 0

The second derivative of f(x) is: 
f''(x) = 2a

Substituting x=2, we get: 
2a = 0

From the second equation, we see that a=0. Substituting this into the first equation, we get: 
b = 0

Therefore, the only choice that satisfies the conditions is D, with a=-1 and b=4.

Provide an Appropriate Response A) a = 1, B = 4 B) a =

Provide an Appropriate Response A) a = 1, B = 4
B) a =