Question

Use the intermediate Value theorem to show that f(x) = x^5 + 7x^2 - 3x+5 has a zero between -2 and -3.

Answer 1

f(-2) = 7
f(-3) = -166
Since f(x) is a polynomial, it is continuous. Consequently, f(x) must pass through every value in between 7 and -166, including 0;hence, f(x) has at least one zero.

Answer 2

Note that for every value in [7, 166] there is a corresponding x that maps to it by virtue of the fact that f(x) is continuous. One of these values in [7,166] is zero and has a value in [-3, -2] that maps to it according to the IMVT