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Vedmedyk [2.9K]
1 year ago
6

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

Mathematics
1 answer:
ANTONII [103]1 year ago
7 0

Answer:

See Below.

Step-by-step 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 <em>x</em> = 1 and <em>x</em> = 2.

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

Evaluate the function at <em>x</em> = 1 and <em>x</em> = 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 <em>x</em> = 1 to <em>x</em> = 2 and is continuous on the interval [1, 2], by the intermediate value theorem, there must exist at least one zero in the interval.

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