Question

find a) any critical values and b) any relative extrema
$t(x)=x^{3}+13x - 1$

a) select the correct choice below and, if necessary, fill in the answer box within your choice
a. the critical value(s) of the function is/are
(use a comma to separate answers as needed )
b. the function has no critical values
b) select the correct choice below and, if necessary, fill in the answer box(es) within your choice
a. the relative minimum point(s) is/are and the relative maximum point(s) is/are
(simplify your answers. type ordered pairs, using integers or fractions. use a comma to separate answers as needed )
b. the relative minimum point(s) is/are and there are no relative maximum points
(simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed )
c. the relative maximum point(s) is/are and there are no relative minimum points
(simplify your answer. type an ordered pair, using integers or fractions. use a comma to separate answers as needed )
d. there are no relative minimum points and there are no relative maximum points

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