find a) any critical values and b) any relative extrema\n$t(x)=x^{3}+13x - 1$\n\n a) select the correct…

find a) any critical values and b) any relative extrema\n$t(x)=x^{3}+13x - 1$\n\n a) select the correct choice below and, if necessary, fill in the answer box within your choice\na. the critical value(s) of the function is/are\n(use a comma to separate answers as needed )\nb. the function has no critical values\nb) select the correct choice below and, if necessary, fill in the answer box(es) within your choice\na. the relative minimum point(s) is/are and the relative maximum point(s) is/are\n(simplify your answers. type ordered pairs, using integers or fractions. use a comma to separate answers as needed )\nb. the relative minimum point(s) is/are and there are no relative maximum points\n(simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed )\nc. the relative maximum point(s) is/are and there are no relative minimum points\n(simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed )\nd. there are no relative minimum points and there are no relative maximum points

find a) any critical values and b) any relative extrema\n$t(x)=x^{3}+13x - 1$\n\n a) select the correct choice below and, if necessary, fill in the answer box within your choice\na. the critical value(s) of the function is/are\n(use a comma to separate answers as needed )\nb. the function has no critical values\nb) select the correct choice below and, if necessary, fill in the answer box(es) within your choice\na. the relative minimum point(s) is/are and the relative maximum point(s) is/are\n(simplify your answers. type ordered pairs, using integers or fractions. use a comma to separate answers as needed )\nb. the relative minimum point(s) is/are and there are no relative maximum points\n(simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed )\nc. the relative maximum point(s) is/are and there are no relative minimum points\n(simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed )\nd. there are no relative minimum points and there are no relative maximum points

Answer

Explanation:

Step1: Find the derivative of (t(x))

The function is (t(x)=x^{3}+13x - 1). Using the power rule ((x^{n})^\prime=nx^{n - 1}), we get (t^\prime(x)=3x^{2}+13).

Step2: Find critical values

Set (t^\prime(x) = 0), so (3x^{2}+13=0). Then (3x^{2}=- 13), and (x^{2}=-\frac{13}{3}). Since the square of a real - number (x) ((x^{2}\geq0) for all real (x)) cannot be negative ((-\frac{13}{3}<0)), there are no real solutions for (x). So, there are no critical values.

Step3: Analyze relative extrema

Since there are no critical values (points where (t^\prime(x) = 0) or (t^\prime(x)) is undefined, and (t^\prime(x)=3x^{2}+13) is defined for all real (x)), there are no relative extrema.

Answer:

a) B. The function has no critical values b) D. There are no relative minimum points and there are no relative maximum points