question 1 of 10, step 1 of 2 consider the function: f(x)=2x³ - 6x² - 90x + 9 step 1 of 2: find the critical…

question 1 of 10, step 1 of 2 consider the function: f(x)=2x³ - 6x² - 90x + 9 step 1 of 2: find the critical values of the function. separate multiple answers with commas. answer how to enter your answer (opens in new window) selecting a radio button will replace the entered answer value(s) with the radio button value. if the radio button is not selected, the entered answer is used. x = none

question 1 of 10, step 1 of 2 consider the function: f(x)=2x³ - 6x² - 90x + 9 step 1 of 2: find the critical values of the function. separate multiple answers with commas. answer how to enter your answer (opens in new window) selecting a radio button will replace the entered answer value(s) with the radio button value. if the radio button is not selected, the entered answer is used. x = none

Answer

Explanation:

Step1: Find the derivative

The derivative of $f(x)=2x^{3}-6x^{2}-90x + 9$ using the power - rule $(x^n)^\prime=nx^{n - 1}$ is $f^\prime(x)=6x^{2}-12x-90$.

Step2: Set the derivative equal to zero

Set $f^\prime(x)=0$, so $6x^{2}-12x - 90=0$. Divide through by 6 to simplify: $x^{2}-2x - 15=0$.

Step3: Solve the quadratic equation

Factor the quadratic equation $x^{2}-2x - 15=(x - 5)(x+3)=0$. Then, using the zero - product property, if $(x - 5)(x + 3)=0$, then $x-5=0$ or $x + 3=0$. Solving these gives $x = 5$ or $x=-3$.

Answer:

$x = 5,-3$