Question

find a) any critical values and b) any relative extrema. t(x)=x³ + 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

Explanation:

Step1: Find the derivative of the function

The derivative of \(t(x)=x^{3}+13x - 1\) using the power rule \((x^{n})^\prime=nx^{n - 1}\) is \(t^\prime(x)=3x^{2}+13\).

Step2: Set the derivative equal to zero and solve for \(x\)

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 \(x\) (i.e., \(x^{2}\)) is non - negative for all real \(x\) (\(x^{2}\geq0\) for \(x\in R\)), there are no real solutions for \(x\) from the equation \(x^{2}=-\frac{13}{3}\).

Answer:

B. The function has no critical values.