Question

Use the Intermediate Value Theorem and a graphing utility to find intervals of length 1 in which the polynomial is guaranteed to have a zero. Use the root feature of a graphing utility to approximate the zeros of the function.
h(x)=x^4-10x^2+2

Answer 1

The intermediate value theorem states that:
"for each value between the least upper bound and greatest lower bound of the image of a continuous function there is a corresponding point in its domain that the function maps to that value".

This implies that when the function evaluated at the limits of an interval have opposing signs, a zero of the function exists between the limits.

By graphing the function, you will find that all four real roots fall between -4 and +4. Use the roots features available in most graphing calculators to find the "exact" roots.

Hint: you can check the results by substituting y=x² which transforms the function to a quadratic function in y that can be solved exactly. x is simply ±√(y).

Answer 2

e^x=x^2