QuestionGiven the function $F(x)=-x^{4}+16 x^{2}+225$ , find if it's even/odd, another local max, and area from $x=-5$ to $x=0$ .

Studdy Solution

STEP 1

Assumptions1. The function is $(x)=-x^{4}+16 x^{}+225$ . We are asked to determine if the function is even, odd, or neither3. We are given that there is a local maximum value of289 at $x=\sqrt{8}$
4. We are asked to determine a second local maximum value5. We are given that the area of the region enclosed by the graph of and the $x$-axis between $x=0$ and $x=5$ is1166.7 square units6. We are asked to determine the area of the region enclosed by the graph of and the $x$ -axis between $x=-5$ and $x=0$ using the result from (a)

STEP 2

(a) To determine if the function is even, odd, or neither, we need to check the following conditions1. A function is even if $(-x) = F(x)$ 2. A function is odd if $(-x) = -(x)$ Let's calculate $(-x)$ and compare it with $(x)$ .
$(-x) = -(-x)^{4}+16 (-x)^{2}+225$

STEP 3

implify $(-x)$ .
$(-x) = -x^{}+16 x^{2}+225$

STEP 4

Since $(-x) = F(x)$ , we can conclude that the function $(x)$ is even.

STEP 5

(b) Since $(x)$ is an even function and it has a local maximum at $x=\sqrt{8}$ , it will also have a local maximum at $x=-\sqrt{8}$ due to symmetry. The value of the function at this point will be the same as at $x=\sqrt{8}$ , which is289.

STEP 6

(c) Since $(x)$ is an even function, the area of the region enclosed by the graph of $$ and the $x$-axis between $x=-5$ and $x=0$ will be the same as the area between $x=0$ and $x=5$, which is1166. square units.
The solutions are(a) The function $(x)$ is even. (b) The second local maximum value is289 at $x=-\sqrt{8}$ . (c) The area of the region enclosed by the graph of $$ and the $x$-axis between $x=-5$ and $x=0$ is1166. square units.

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QuestionGiven the function $F(x)=-x^{4}+16 x^{2}+225$ , find if it's even/odd, another local max, and area from $x=-5$ to $x=0$ .

Studdy Solution

STEP 1

STEP 2

(a) To determine if the function is even, odd, or neither, we need to check the following conditions1. A function is even if $(-x) = F(x)$ 2. A function is odd if $(-x) = -(x)$ Let's calculate $(-x)$ and compare it with $(x)$ .
$(-x) = -(-x)^{4}+16 (-x)^{2}+225$

STEP 3

implify $(-x)$ .
$(-x) = -x^{}+16 x^{2}+225$

STEP 4

Since $(-x) = F(x)$ , we can conclude that the function $(x)$ is even.

STEP 5

(b) Since $(x)$ is an even function and it has a local maximum at $x=\sqrt{8}$ , it will also have a local maximum at $x=-\sqrt{8}$ due to symmetry. The value of the function at this point will be the same as at $x=\sqrt{8}$ , which is289.

STEP 6

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QuestionGiven the function F(x)=−x4+16x2+225F(x)=-x^{4}+16 x^{2}+225F(x)=−x4+16x2+225, find if it's even/odd, another local max, and area from x=−5x=-5x=−5 to x=0x=0x=0.

Studdy Solution

STEP 1

STEP 2

STEP 3

implify (−x)(-x)(−x).(−x)=−x+16x2+225(-x) = -x^{}+16 x^{2}+225(−x)=−x+16x2+225

STEP 4

Since (−x)=F(x)(-x) = F(x)(−x)=F(x), we can conclude that the function (x)(x)(x) is even.

STEP 5

(b) Since (x)(x)(x) is an even function and it has a local maximum at x=8x=\sqrt{8}x=8​, it will also have a local maximum at x=−8x=-\sqrt{8}x=−8​ due to symmetry. The value of the function at this point will be the same as at x=8x=\sqrt{8}x=8​, which is289.

STEP 6

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QuestionGiven the function $F(x)=-x^{4}+16 x^{2}+225$ , find if it's even/odd, another local max, and area from $x=-5$ to $x=0$ .

implify $(-x)$ .
$(-x) = -x^{}+16 x^{2}+225$

Since $(-x) = F(x)$ , we can conclude that the function $(x)$ is even.

(b) Since $(x)$ is an even function and it has a local maximum at $x=\sqrt{8}$ , it will also have a local maximum at $x=-\sqrt{8}$ due to symmetry. The value of the function at this point will be the same as at $x=\sqrt{8}$ , which is289.