QuestionFor the function , find if is even/odd, a second local max, and area from to .
Studdy Solution
STEP 1
Assumptions1. The function is
. We need to determine if the function is even, odd, or neither3. There is a local maximum value of196 at
4. We need to find a second local maximum value5. The area of the region enclosed by the graph of and the $x$-axis between $x=0$ and $x=4$ is648.5 square units6. We need to find the area of the region enclosed by the graph of and the -axis between and
STEP 2
To determine if a function is even, odd, or neither, we can use the following properties- A function is even if for all in the domain of
- A function $(x)$ is odd if $(-x) = -(x)$ for all $x$ in the domain of
Let's substitute into the function and simplify.
STEP 3
implify the equation.
STEP 4
Since , we can conclude that the function is even.
STEP 5
The function has a local maximum at . Since is an even function, it is symmetric about the y-axis. Therefore, there is another local maximum at .
STEP 6
To find the value of the second local maximum, we substitute into the function.
STEP 7
implify the equation.
So, the second local maximum value is196.
STEP 8
The area of the region enclosed by the graph of and the $x$-axis between $x=0$ and $x=4$ is given as648.5 square units. Since $(x)$ is an even function, the area of the region enclosed by the graph of and the -axis between and is also648.5 square units.
Was this helpful?
