Question

3x3+12x-10 find the vertex.

Answer 1

`\begin{gathered} -3x^3+12x-10 \\ \text{rewrite the function as:} \\ x(12-3x^2)-10 \end{gathered}`
`\begin{gathered} \text{let:} \\ f(x)=-3x^3+12x-10 \\ f^(\prime)(x)=-9x^2+12 \end{gathered}`

Let's find the critical points:

`\begin{gathered} f^(\prime)(x)=0 \\ -9x^2+12=0 \\ x^2=(12)/(9) \\ x=\pm\frac{2\sqrt[]{3}}{3}\approx\pm1.155 \end{gathered}`

Now:

`\begin{gathered} \text{for:} \\ x<-1.155 \\ f^(\prime)(-2)=-24<0 \\ \text{for:} \\ x>-1.155 \\ f^(\prime)(-1)=3>0 \\ \text{for:} \\ x<1.155 \\ f^(\prime)(1)=3>0 \\ \text{for:} \\ x>1.155 \\ f(2)=-12<0 \end{gathered}`

Therefore, the vertex of the function are:

`\begin{gathered} x=-1.155,f(-1.155)=-19.238 \\ \text{and} \\ x=1.155,f(1.155)=-0.762 \end{gathered}`