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