Question
Consider the function
.
a. Determine, without graphing. whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function’s domain and its range.
a. The function has a
a. Determine, without graphing. whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function’s domain and its range.
a. The function has a
Ask by Gibson Pritchard. in the United States
Nov 02,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
a. The function has a maximum value.
b. The maximum value is 45, occurring at .
c. The domain is and the range is
.
b. The maximum value is 45, occurring at
c. The domain is
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The function has a maximum value because the coefficient of
is negative (-2), indicating that the parabola opens downward. This means the vertex of the parabola will represent the maximum point.
To find the maximum value, we can use the vertex formula
. Here,
and
. Plugging in these values gives
. Now, substituting
back into the function,
. Thus, the maximum value is 50 at
.
The function’s domain is all real numbers, denoted as
. For the range, since it opens downward with a maximum of 50, the range is
.