Let's break this down step by step as requested, with explanations at each step. --- ### **Step 1: Analyze the Table and Symmetry (Even Function)** - **Even and continuous:** The function \( f(x) \) is **even**, so \( f(-x) = f(x) \) for all \( x \). Its graph is symmetric about the y-axis. - **At \( x = 0 \):** - \( f(0) = 1 \) - \( f'(0) \) and \( f''(0) \) are **undefined** (so sharp point or cusp at \( x=0 \)). - **For \( 0 < x < 2 \):** - \( f(x) \) is positive and decreasing (since \( f'(x) < 0 \)), and concave up (since \( f''(x) > 0 \)). - **At \( x = 2 \):** - \( f(2) = 0 \) (x-intercept), \( f'(2) = 0 \), \( f''(2) = 0 \) (possible inflection or stationary point). - **For \( 2 < x < 4 \):** - \( f(x) \) is negative and decreasing (since \( f'(x) < 0 \)), and concave down (since \( f''(x) < 0 \)). - **At \( x = 4 \):** - \( f(4) = -1 \), \( f'(4) \) is undefined (another sharp point or cusp), \( f''(4) \) is undefined. - **For \( 4 < x < 5 \):** - \( f(x) \) is negative and increasing (since \( f'(x) > 0 \)), and concave down (since \( f''(x) < 0 \)). **Explanation:** By reading the table and considering the even property, we can deduce the overall "shape" of the graph. The function starts at \( (0,1) \), decreases (concave up) to \( (2,0) \), continues decreasing (concave down) to \( (4,-1) \), then increases (still concave down) toward \( x = 5 \). The pattern is mirrored for \( x < 0 \). --- ### **Step 2: Sketch the Graph Based on the Analysis** - **From \( x = 0 \) to \( x = 2 \):** - Start at \( (0,1) \) with a sharp point/cusp. - Curve downward (concave up) to \( (2,0) \). - **From \( x = 2 \) to \( x = 4 \):** - Continue decreasing (now concave down) to \( (4,-1) \). - **From \( x = 4 \) to \( x = 5 \):** - Increase (still concave down) but stay negative. - **Mirror this on the left side (\( x < 0 \)) due to evenness.** **Explanation:** The graph should have a cusp or sharp point at \( x = 0 \) and at \( x = \pm4 \). It should be symmetric about the y-axis, cross the x-axis at \( x = \pm2 \), and reach minimums at \( x = \pm4 \) with value \(-1\), then rise a bit as \( x \) approaches \( \pm5 \). --- ### **Final Answer:** **Sketch the graph as follows:** - Symmetric about the y-axis. - Starts at \( (0,1) \) with a cusp. - Decreases (concave up) to \( (2,0) \), then decreases (concave down) to \( (4,-1) \) with a cusp at \( x=4 \). - Then increases (concave down) but remains negative as \( x \) approaches 5. - Mirror this for negative \( x \). **The graph should look like a sharp "W" shape, symmetric about the y-axis, with cusps at \( x = 0 \) and \( x = \pm4 \), crossing the x-axis at \( x = \pm2 \).**