Determine if the statement is true or false. If a statement is false, explain why. Suppose that \( f(x) \) is a polynomial, and that \( a \) and \( b \) are real numbers where \( a

Ask by Powell Clarke. in the United States

Mar 16,2025

False. The given statement claims that if \( f(a)<0 \) and \( f(b)<0 \), then \( f(x) \) has no real zeros on the interval \( [a, b] \). However, this is not necessarily true. A polynomial can dip below the x-axis and then return below it again without necessarily crossing the x-axis to create real zeros. For instance, the polynomial \( f(x) = (x-1)^2 - 1 \) is negative at both ends of the interval \( [0, 2] \) but never crosses the x-axis; thus, it has no real zeros in that interval. So, the correct conclusion would be that just having \( f(a)<0 \) and \( f(b)<0 \) does not guarantee the absence of real zeros between \( a \) and \( b \).