Question

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

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 \(x=\frac{1}{3},\frac{3}{2}\)