find the critical points of the following function.\n\n$f(x)=4x^{3}+\\frac{17}{2}x^{2}+6x$\n\nselect the…

find the critical points of the following function.\n\n$f(x)=4x^{3}+\\frac{17}{2}x^{2}+6x$\n\nselect the correct choice and, if necessary, fill in the answer box to complete your choice.\n\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.\n\n$f(x)=4x^{3}+\\frac{17}{2}x^{2}+6x$\n\nselect the correct choice and, if necessary, fill in the answer box to complete your choice.\n\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 = 4x^{3}+\frac{17}{2}x^{2}+6x), the derivative (y^\prime=f^\prime(x)=4\times3x^{2}+\frac{17}{2}\times2x + 6). Simplify to get (f^\prime(x)=12x^{2}+17x + 6).

Step2: Find the roots of the derivative

Set (f^\prime(x)=0), so (12x^{2}+17x + 6 = 0). Use the quadratic formula (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}) for (ax^{2}+bx + c=0). Here (a = 12), (b=17), (c = 6). First, calculate the discriminant (\Delta=b^{2}-4ac=(17)^{2}-4\times12\times6=289 - 288=1). Then (x=\frac{-17\pm\sqrt{1}}{2\times12}=\frac{-17\pm1}{24}). For the plus - case: (x=\frac{-17 + 1}{24}=\frac{-16}{24}=-\frac{2}{3}). For the minus - case: (x=\frac{-17-1}{24}=\frac{-18}{24}=-\frac{3}{4}).

Answer:

A. The critical point(s) occur(s) at (x=-\frac{3}{4},-\frac{2}{3})