Question

A) Use interval notation to indicate where f(x) is increasing B) use interval notation to indicate where f(x) is decreasingC) List the X values of all local maximum of FD) List the X values of all local minimum of F

Answer 1

`\begin{gathered} a)\text{ }(\sqrt{(7)/(3)},\sqrt{(7)/(3)) \\ b)(-\infty,\sqrt{(7)/(3)})\cup(\sqrt{(7)/(3)},\:\infty) \\ c)\left(\sqrt{(7)/(3)},\:8+7\sqrt{(7)/(3)}-\left((7)/(3)\right)^{(3)/(2)}\right) \\ d)\left(-\sqrt{(7)/(3)},\:8-7\sqrt{(7)/(3)}+\left((7)/(3)\right)^{(3)/(2)}\right) \end{gathered}`

1) In this question we'll need to find the monotone intervals for a, and b and for c and d perform some derivative tests so that we can tell, algebraically, how the function behaves.

a) Increasing

We need to find the monotone intervals, taking the first derivative

`\begin{gathered} (d)/(dx)\left(8+7x-x^3\right) \\ (d)/(dx)\left(8\right)+(d)/(dx)\left(7x\right)-(d)/(dx)\left(x^3\right) \\ 0+7-3x \\ f^(\prime)\left(x\right)=-3x+7 \end{gathered}`

Since we want to know the intervals in which this function is increasing then we need to write out the following inequality: f'(x)>0

b)Decreasing

From the work in the previous section, we can tell the following intervals indicate where the function decreases

`\begin{gathered} -3x^2+7<0 \\ -3x^2+7-7>0-7 \\ -3x^2>-7 \\ 3x^2<7 \\ x<-\sqrt{(7)/(3)}\quad \mathrm{or}\quad \:x>\sqrt{(7)/(3)} \\ \end{gathered}`

As for intervals, we can tell, by combining them with the Domain:

Now let's plug into that the x-coordinate we found in the previous section:

`\begin{gathered} f\mleft(x\mright)=8+7x-x^3 \\ f\mleft(\sqrt{(7)/(3)}\mright)=8+7\left(\sqrt{(7)/(3)}\right?-\left(\sqrt{(7)/(3)}\right?^3 \\ \left(\sqrt{(7)/(3)},\:8+7\sqrt{(7)/(3)}-\left((7)/(3)\right)^{(3)/(2)}\right) \end{gathered}`

Since it is an irrational number we can leave it there in radical form.

d) Local Minimum.

From the previous work on top, we know the point and the coordinates. So combining the intervals we can tell that this is the Minimum :

`\left(-\sqrt{(7)/(3)},\:8-7\sqrt{(7)/(3)}+\left((7)/(3)\right)^{(3)/(2)}\right)`