find the x - values of all points where the function has any relative extrema. find the value(s) of any…

find the x - values of all points where the function has any relative extrema. find the value(s) of any relative extrema.\n\nf(x)=x² + x + 1\n\nselect the correct choice below and, if necessary, fill in any answer boxes within your choice.\n\na. there are no relative maxima. the function has a relative minimum of □ at x = □.\n(use a comma to separate answers as needed.)\nb. the function has a relative maximum of □ at x = □ and a relative minimum of □ at x = □.\n(use a comma to separate answers as needed.)\nc. there are no relative minima. the function has a relative maximum of □ at x = □.\n(use a comma to separate answers as needed.)\nd. 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.\n\nf(x)=x² + x + 1\n\nselect the correct choice below and, if necessary, fill in any answer boxes within your choice.\n\na. there are no relative maxima. the function has a relative minimum of □ at x = □.\n(use a comma to separate answers as needed.)\nb. the function has a relative maximum of □ at x = □ and a relative minimum of □ at x = □.\n(use a comma to separate answers as needed.)\nc. there are no relative minima. the function has a relative maximum of □ at x = □.\n(use a comma to separate answers as needed.)\nd. there are no relative extrema.

Answer

Explanation:

Step1: Find the first - derivative of (f(x))

Given (f(x)=x^{2}+x + 1). Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (f^\prime(x)=(x^{2})^\prime+(x)^\prime+(1)^\prime). (f^\prime(x)=2x + 1).

Step2: Find the critical points

Set (f^\prime(x)=0), so (2x+1 = 0). Solve for (x): (2x=-1), then (x=-\frac{1}{2}).

Step3: Find the second - derivative of (f(x))

Differentiate (f^\prime(x)=2x + 1) with respect to (x). Using the power rule, (f^{\prime\prime}(x)=(2x)^\prime+(1)^\prime). (f^{\prime\prime}(x)=2).

Step4: Use the second - derivative test

Since (f^{\prime\prime}(x)=2>0) for all (x), when (x =-\frac{1}{2}), the function (f(x)) has a relative minimum. There are no relative maxima.

Answer:

C. There are no relative maxima. The function has a relative minimum of (x =-\frac{1}{2}).