Question

Find all critical points of the function f(x) = x^3 + 5x^2 - 7x - 3. If there is more than one critical point, list them in descending order and separate them by commas.The critical point(s) is(are) =

Answer 1

Given the following question:

$f\mleft(x\mright)=x^3+5x^2-7x-3$$\begin{gathered} f(x)=x^{3}+5x^{2}-7x-3 \\ x^3=3x^2 \\ x^2=2x \\ 5x\times2x=10x \\ x=1 \\ 7x=7\left(1\right)=7\times1=7 \\ 3x^2+10x-7 \\ \text{ To find the critical points} \\ 3x^2+10x-7=0 \\ -\frac{5}{3}+\frac{\sqrt{46}}{3} \\ -\frac{\sqrt{46}}{3}-\frac{5}{3} \\ \text{ The critical points are:} \\ -\frac{5-\sqrt{46}}{3},\frac{484-92\sqrt{46}}{27} \\ -\frac{5+\sqrt{46}}{3},\frac{484+92\sqrt{46}}{27} \\ \end{gathered}$