Question

Show that the function has at least one zero between x=1 and x=2

Answer 1

See Below.

Explanation:

We are given the function:

`\displaystyle f(x) = 7x^5-9x^4-x^2`

And we want to show that it has at least one zero between x = 1 and x = 2.

Because the function is a polynomial, it is everywhere continuous.

Evaluate the function at x = 1 and x = 2:

`\displaystyle \begin{aligned} f(1) & = 7(1)^5 - 9(1)^4 - (1)^2 \\ \\ & = -3\end{aligned}`

And:

`\displaystyle \begin{aligned} f(2) & = 7(2)^5 - 9(2)^4 - (2)^2 \\ \\ & = 76 \end{aligned}`

Therefore, because the function changes signs from x = 1 to x = 2 and is continuous on the interval [1, 2], by the intermediate value theorem, there must exist at least one zero in the interval.