Question

the function $f(x)=2x^{3}-24x^{2}+90x - 2$ has two critical values. the smaller one equals 3 and the larger one equals 5

Explanation:

Step1: Find the derivative

The derivative of $f(x)=2x^{3}-24x^{2}+90x - 2$ using the power - rule $(x^n)'=nx^{n - 1}$ is $f'(x)=6x^{2}-48x + 90$.

Step2: Set the derivative equal to zero

Set $f'(x)=0$, so $6x^{2}-48x + 90 = 0$. Divide through by 6 to get $x^{2}-8x + 15=0$.

Step3: Solve the quadratic equation

Factor the quadratic equation $x^{2}-8x + 15=(x - 3)(x - 5)=0$. Then, by the zero - product property, $x=3$ or $x = 5$.

Answer:

The smaller critical value is 3 and the larger critical value is 5.