Question

the function f(x)=2x^3 - 24x^2 + 90x - 2 has two critical values. the smaller one equals and the larger one equals submit answer next item

Explanation:

Step1: Find the derivative of the function

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

Step2: Set the derivative equal to zero

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

Step3: Solve the quadratic equation

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

Answer:

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