Let's break down the information step by step and sketch the graph based on the given properties. ### **Step 1: Understand the Symmetry** - The function \( f \) is **even**, so \( f(-x) = f(x) \). This means the graph is symmetric about the y-axis. --- ### **Step 2: Analyze the Table (on \( [0, 5] \))** #### **At \( x = 0 \):** - \( f(0) = 1 \) - \( f'(0) \) is **undefined** - \( f''(0) \) is **undefined** #### **On \( 0 < x < 2 \):** - \( f(x) \) is **positive** - \( f'(x) \) is **negative**: Decreasing - \( f''(x) \) is **positive**: Concave up #### **At \( x = 2 \):** - \( f(2) = 0 \) - \( f'(2) = 0 \) - \( f''(2) = 0 \) #### **On \( 2 < x < 4 \):** - \( f(x) \) is **negative** - \( f'(x) \) is **negative**: Decreasing - \( f''(x) \) is **negative**: Concave down #### **At \( x = 4 \):** - \( f(4) = -1 \) - \( f'(4) \) is **undefined** - \( f''(4) \) is **undefined** #### **On \( 4 < x < 5 \):** - \( f(x) \) is **negative** - \( f'(x) \) is **positive**: Increasing - \( f''(x) \) is **negative**: Concave down --- ### **Step 3: Key Points and Behavior** - **At \( x = 0 \), \( f(0) = 1 \); sharp point or cusp (since \( f'(0) \) undefined).** - **At \( x = 2 \), \( f(2) = 0 \); possible inflection point (since all derivatives are zero).** - **At \( x = 4 \), \( f(4) = -1 \); cusp or sharp point.** - **On \( 0 < x < 2 \), \( f(x) \) is positive, decreasing, concave up.** - **On \( 2 < x < 4 \), \( f(x) \) is negative, decreasing, concave down.** - **On \( 4 < x < 5 \), \( f(x) \) is negative, increasing, concave down.** --- ### **Step 4: Sketch on \( [0, 5] \)** 1. Start at \( (0, 1) \) with a cusp. 2. Decrease (concave up) until \( (2, 0) \). 3. Continue decreasing (now concave down) until \( (4, -1) \) with a cusp. 4. From \( (4, -1) \), start increasing (still concave down) toward \( x = 5 \), but stay negative. --- ### **Step 5: Symmetry** - Reflect the graph across the y-axis for \( x < 0 \) because \( f \) is even. --- ### **Step 6: Final Sketch** Below is a rough sketch description: ``` y ^ | /\ | / \ | / \ | / \ | / \ | / \ |--/-------------\-------> x | / \ |/ \ ``` - The peak at \( (0,1) \) is a cusp. - The graph decreases (concave up) to \( (2,0) \) (inflection point). - Continues down (concave down) to \( (4,-1) \) (cusp). - From \( (4,-1) \), increases but stays negative as \( x \to 5 \). - Reflect this for \( x < 0 \) due to evenness. --- ### **Summary Table of Key Features** | Point | Value | Behavior | |-----------|------------|-------------------------| | \( x=0 \) | \( f=1 \) | Cusp, start | | \( x=2 \) | \( f=0 \) | Inflection, all derivs 0| | \( x=4 \) | \( f=-1 \) | Cusp | --- ### **Final Answer:** Draw a graph that is symmetric about the y-axis, starts at \( (0,1) \) with a cusp, decreases concave up to \( (2,0) \), continues down concave down to \( (4,-1) \) (cusp), then increases (concave down) but remains negative as \( x \to 5 \). Reflect for \( x < 0 \). Would you like a hand-drawn/sketched image of the graph as well?