Question

Use the intermediate value theorem to determine whether f(x)=8x^4-9x^2-9 has a real zero between 1 and 2

Answer 1

There is a real zero in this interval

Explanation:

We have the following function:

`f\mleft(x\mright)=8x^4-9x^2-9`

To determine if there is a real zero between 1 and 2, we must evaluate the function at these points, if there is a change from positive to negative or vice versa, by the intermediate value theorem we can say that it has a real zero in that interval.

`\begin{gathered} f(1)=8\left(1\right)^4-9\left(1\right)^2-9 \\ \\ f(1)=8\cdot1-9\cdot1-9 \\ \\ f(1)=8-9-9 \\ \\ f(1)=-10 \\ \\ \\ f(2)=8\left(2\right)^4-9\left(2\right)^2-9 \\ \\ f(2)=8\cdot16-9\cdot4-9 \\ \\ f(2)=128-36-9 \\ \\ f(2)=83 \end{gathered}`

We can observe that it goes from a negative value to a positive value in this small interval, which by means of the theorem we can say that if there is a real zero in this interval