Let's break down the given problem step by step. --- ### **Problem Statement** Suppose \( f \) is continuous on \([a, b]\). #### (a) If \( f(a) < 0 \) and \( f(b) > 0 \), what can you say about \( f \)? #### (b) If \( f(a) > 0 \) and \( f(b) = 0 \), what can you say about \( f \)? --- ### **Step-by-step Solution** #### **Step 1: Understanding the Theorem** The **Intermediate Value Theorem** states: > If a function \( f \) is continuous on a closed interval \([a, b]\) and \(N\) is any number between \(f(a)\) and \(f(b)\), then there exists a number \(c\) in \((a, b)\) such that \(f(c) = N\). --- #### **Part (a): \( f(a) < 0 \) and \( f(b) > 0 \)** - \(f(a)\) is negative and \(f(b)\) is positive. - 0 is a number between \(f(a)\) and \(f(b)\). **By the Intermediate Value Theorem:** - There exists some \(c\) in \((a, b)\) such that \(f(c) = 0\). **Explanation:** Because \(f\) is continuous, the function must cross the x-axis somewhere between \(a\) and \(b\). --- #### **Part (b): \( f(a) > 0 \) and \( f(b) = 0 \)** - \(f(a)\) is positive and \(f(b)\) is zero. - 0 is between \(f(a)\) and \(f(b)\) (since \(f(b) = 0\)). **By the Intermediate Value Theorem:** - There exists some \(c\) in \([a, b]\) such that \(f(c) = 0\). **But since \(f(b) = 0\), we know that at \(b\), \(f(b) = 0\).** So, **at least at \(b\)**, \(f\) is zero. There may not be another zero in \((a, b)\) unless the function crosses zero before \(b\). --- ### **Final Answers** **(a)** If \( f(a) < 0 \) and \( f(b) > 0 \), then **there must be some \( c \) in \( (a, b) \) such that \( f(c) = 0 \)**. **(b)** If \( f(a) > 0 \) and \( f(b) = 0 \), then **\( f(b) = 0 \)**, so at least at \( b \), the function is zero; the Intermediate Value Theorem does not guarantee any zero in \( (a, b) \), only at \( b \). --- #### **Summary Table** | Condition | What can you say? | |--------------------------------------------|-----------------------------------------------------| | \( f(a) < 0 \), \( f(b) > 0 \) | There is \( c \in (a, b) \) with \( f(c) = 0 \) | | \( f(a) > 0 \), \( f(b) = 0 \) | \( f(b) = 0 \), possibly only at \( b \) |