Let's solve step by step: ### **Step 1: Interpreting the Sign Charts** #### **For \( f(x) \):** - **Intervals:** - \( (-\infty, -2) \): \( f(x) \) is decreasing - \( (-2, 0) \): \( f(x) \) is increasing - \( (0, 1) \): \( f(x) \) is increasing - \( (1, 2) \): \( f(x) \) is decreasing - \( (2, \infty) \): \( f(x) \) is increasing #### **For \( f'(x) \):** - **DNE at \( x = -1 \), \( x = 1 \)** - **Intervals:** - \( (-\infty, -1) \): \( f'(x) > 0 \) - \( (-1, 0) \): \( f'(x) < 0 \) - \( (0, 1) \): \( f'(x) > 0 \) - \( (1, \infty) \): \( f'(x) < 0 \) #### **For \( f''(x) \):** - **DNE at \( x = -1, 1 \)** - **Intervals:** - \( (-\infty, -1) \): \( f''(x) > 0 \) (concave up) - \( (-1, 1) \): \( f''(x) < 0 \) (concave down) - \( (1, \infty) \): \( f''(x) > 0 \) (concave up) --- ### **Step 2: Limits and Given Point** - \( \lim_{x \to -\infty} f(x) = -3 \) - \( \lim_{x \to \infty} f(x) = \infty \) - \( f(-1) = 1 \) --- ### **Step 3: Key Features** #### **Critical Points and Behavior:** - At \( x = -2 \), \( f(x) \) switches from decreasing to increasing → local **min**. - At \( x = 0 \), \( f(x) \) switches from increasing to increasing (no extremum), but \( f'(x) \) switches from negative to positive, so **local min** at \( x = 0 \). - At \( x = 1 \), \( f(x) \) switches from increasing to decreasing → local **max**. - At \( x = 2 \), \( f(x) \) switches from decreasing to increasing → local **min**. #### **Inflection Points:** - At \( x = -1 \): \( f''(x) \) DNE, possible cusp/vertical tangent; \( f'(x) \) switches from positive to negative → local **max** at \( x = -1 \). - At \( x = 1 \): \( f''(x) \) DNE, possible cusp/vertical tangent; \( f'(x) \) switches from positive to negative → local **max** at \( x = 1 \). --- ### **Step 4: Sketch the Graph** #### **Mark Key Points:** - Start at \( y = -3 \) as \( x \to -\infty \). - Pass through \( (-1, 1) \). - Head up to infinity as \( x \to \infty \). #### **Behavior:** - **From \( x = -\infty \) to \( x = -2 \):** Decreasing and concave up. - **At \( x = -2 \):** Local min. - **From \( x = -2 \) to \( x = -1 \):** Increasing and concave up. - **At \( x = -1 \):** Sharp local max at \( ( -1, 1 ) \). - **From \( x = -1 \) to \( x = 0 \):** Decreasing and concave down. - **At \( x = 0 \):** Local min. - **From \( x = 0 \) to \( x = 1 \):** Increasing and concave down. - **At \( x = 1 \):** Sharp local max. - **From \( x = 1 \) to \( x = 2 \):** Decreasing and concave up. - **At \( x = 2 \):** Local min. - **From \( x = 2 \) onward:** Increasing, concave up, heads to infinity. --- ### **Step 5: Draw the Graph** I'll describe the sketch: - Start at left at \( y = -3 \), decreasing, concave up, hit a min at \( x = -2 \). - Increase, hit a sharp max at \( ( -1, 1 ) \). - Decrease (concave down), hit a min at \( x = 0 \). - Increase (concave down), hit a sharp max at \( x = 1 \). - Decrease (concave up), hit a min at \( x = 2 \). - Increase (concave up), go to infinity as \( x \to \infty \). --- ### **Final Answer:** #### **(See the sketch below; you can draw this on the axes provided.)** ``` | ∞ | / | / | /-------/ | / | /-------/ | / | /-----/ | / -3 ___/________________________ | -2 -1 0 1 2 ``` - The graph has local mins at \( x = -2, 0, 2 \). - Sharp local maxima at \( x = -1 \) (with \( f(-1) = 1 \)), and at \( x = 1 \). - The function approaches \( y = -3 \) as \( x \to -\infty \) and goes to infinity as \( x \to \infty \). --- If you want a more precise sketch, let me know and I can plot it for you!