find the critical points of the following function.\nf(x)=2x^{3}-\\frac{11}{2}x^{2}+3x\nselect the correct…

find the critical points of the following function.\nf(x)=2x^{3}-\\frac{11}{2}x^{2}+3x\nselect the correct choice and, if necessary, fill in the answer box to complete your choice.\na. the critical point(s) occur(s) at x=\n(simplify your answer. use a comma to separate answers as needed.)\nb. there are no critical points.

find the critical points of the following function.\nf(x)=2x^{3}-\\frac{11}{2}x^{2}+3x\nselect the correct choice and, if necessary, fill in the answer box to complete your choice.\na. the critical point(s) occur(s) at x=\n(simplify your answer. use a comma to separate answers as needed.)\nb. there are no critical points.

Answer

Explanation:

Step1: Differentiate the function

Using the power rule ((x^n)^\prime = nx^{n - 1}), for (y = 2x^{3}-\frac{11}{2}x^{2}+3x), we have (y^\prime=f^\prime(x)=2\times3x^{2}-\frac{11}{2}\times2x + 3). Simplify to get (f^\prime(x)=6x^{2}-11x + 3).

Step2: Find the roots of the derivative

Set (f^\prime(x)=0), so (6x^{2}-11x + 3 = 0). Use the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}) for (ax^{2}+bx + c=0). Here (a = 6), (b=-11), (c = 3). First, calculate the discriminant (\Delta=b^{2}-4ac=(-11)^{2}-4\times6\times3=121 - 72=49). Then (x=\frac{-(-11)\pm\sqrt{49}}{2\times6}=\frac{11\pm7}{12}). For the plus - sign: (x=\frac{11 + 7}{12}=\frac{18}{12}=\frac{3}{2}). For the minus - sign: (x=\frac{11-7}{12}=\frac{4}{12}=\frac{1}{3}).

Answer:

A. The critical point(s) occur(s) at (x=\frac{1}{3},\frac{3}{2})