Questions: Find the x-values of all points where the function has any relative extrema. Find the value(s) of any relative extrema. f(x)=-x^2+5x-2 Select the correct choice below and, if necessary, fill in any answer boxes within your choice. A. The function has a relative maximum of at x= and a relative minimum of at x= 17. (Use a comma to separate answers as needed.) B. There are no relative maxima. The function has a relative minimum of at x= (Use a comma to separate answers as needed.) C. There are no relative minima. The function has a relative maximum of at x= 17. (Use a comma to separate answers as needed.) D. There are no relative extrema.

Find the x-values of all points where the function has any relative extrema. Find the value(s) of any relative extrema. f(x)=-x^2+5x-2 Select the correct choice below and, if necessary, fill in any answer boxes within your choice. A. The function has a relative maximum of at x= and a relative minimum of at x= 17. (Use a comma to separate answers as needed.) B. There are no relative maxima. The function has a relative minimum of at x= (Use a comma to separate answers as needed.) C. There are no relative minima. The function has a relative maximum of at x= 17. (Use a comma to separate answers as needed.) D. There are no relative extrema.
Transcript text: Find the $x$-values of all points where the function has any relative extrema. Find the value(s) of any relative extrema. \[ f(x)=-x^{2}+5 x-2 \] Select the correct choice below and, if necessary, fill in any answer boxes within your choice. A. The function has a relative maximum of $\square$ at $x=$ $\square$ and a relative minimum of $\square$ at $x=$ $\square$ 17. (Use a comma to separate answers as needed.) B. There are no relative maxima. The function has a relative minimum of $\square$ at $x=$ $\square$ (Use a comma to separate answers as needed.) C. There are no relative minima. The function has a relative maximum of $\square$ at $x=$ $\square$ 17. (Use a comma to separate answers as needed.) D. There are no relative extrema.
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Solution

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Solution Steps

Step 1: Find the derivative of $f(x)$

The derivative of $f(x)$ is $f'(x) = 5 - 2 x$.

Step 2: Solve $f'(x) = 0$

The critical points are found by solving $f'(x) = 0$. The solutions are \left\{\frac{5}{2}\right\}.

Step 3: Determine the nature of each critical point

At $x = \frac{5}{2}$, the function has a relative maximum.

Step 4: Find the value of $f(x)$ at each relative extremum

At $x = \frac{5}{2}$, $f(x) = 4.25$.

Final Answer:

The function has a relative maximum at $x = \frac{5}{2}$, with $f(x) = 4.25$.

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