find the critical points of the following function. f(x)=2x³ + 11/2x² - 10x select the correct choice below…

find the critical points of the following function. f(x)=2x³ + 11/2x² - 10x select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) occur(s) at x = (use a comma to separate answers as needed.) b. there are no critical points.

find the critical points of the following function. f(x)=2x³ + 11/2x² - 10x select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) occur(s) at x = (use a comma to separate answers as needed.) b. there are no critical points.

Answer

Explanation:

Step1: Find the derivative

Differentiate $f(x)=2x^{3}+\frac{11}{2}x^{2}-10x$ using power - rule. The derivative $f'(x)=6x^{2}+11x - 10$.

Step2: Set the derivative equal to zero

We have the quadratic equation $6x^{2}+11x - 10 = 0$. For a quadratic equation $ax^{2}+bx + c=0$ ($a = 6$, $b = 11$, $c=-10$), we use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(11)^{2}-4\times6\times(-10)=121 + 240=361$. Then, $x=\frac{-11\pm\sqrt{361}}{2\times6}=\frac{-11\pm19}{12}$.

Step3: Solve for x

For the plus - sign: $x=\frac{-11 + 19}{12}=\frac{8}{12}=\frac{2}{3}$. For the minus - sign: $x=\frac{-11-19}{12}=\frac{-30}{12}=-\frac{5}{2}$.

Answer:

A. The critical point(s) occur(s) at $x=-\frac{5}{2},\frac{2}{3}$