Question

Use the Intermediate Value Theorem to show that the polynomial function has a zero in the given interval.

Answer 1

Given:

$f(x)=10x^4-4x^2+5x-1;\lbrack-2,0\rbrack$

Using the intermediate value theorem,

$\begin{gathered} f(x)=10x^4-4x^2+5x-1 \\ f(-2)=10(-2)^4-4(-2)^2+5(-2)-1 \\ f(-2)=160-16-10-1=133 \\ \text{and} \\ f(0)=10(0)^4-4(0)^2+5(0)-1=-1 \end{gathered}$

So, we have find value c between [-2,0].

$\begin{gathered} f(x)=0 \\ 10x^4-4x^2+5x-1=0 \\ \Rightarrow x=-1\text{ it satisfies the equation} \\ \text{Also, -1}\in\lbrack-2,0\rbrack \end{gathered}$

It shows that, the above polynomial function has zero in the given interval.

Also, the value of f(-2) = 133