It looks like the image you uploaded contains only the questions, but **not the contour plot** itself. In order to answer these questions, I need to see the contour plot, since all the questions refer to points and changes in the plot. ### What to do next: - **Please upload or attach the contour plot** for the function \( f(x, y) \). - Once I have the contour plot, I can answer each question step by step, with explanations. --- ### For your reference, here's how to approach these questions (once the contour plot is available): #### (a) Is \( f_x(2, 1) \) positive or negative? - **Look at how the function changes as you move in the \( x \)-direction** at (2,1). - If the contour values **increase as \( x \) increases**, \( f_x > 0 \). - If the values **decrease**, \( f_x < 0 \). #### (b) Is \( f_y(2, 1) \) positive or negative? - **Look at how the function changes as you move in the \( y \)-direction** at (2,1). - Same logic as above. #### (c) What can you say about the partial derivatives at (1,2)? - If the contours are **parallel or perpendicular** to \( x \)- or \( y \)-axis, or if (1,2) is on a peak/valley/saddle. #### (d) Locate a point where \( f_x(x, y) = 0 \). - Find a place where the contours are **vertical** (moving in \( x \)-direction doesn't change function value). #### (e) Locate a point where \( f_x(x, y) < 0 \). - Find a place where the contours **decrease** as you move to the right. --- **Please upload the contour plot so I can give you detailed answers!**

Certainly! Since the contour plot itself is essential for precise analysis, I will assume a typical scenario based on common features of contour plots and the questions asked. However, to provide accurate step-by-step solutions with calculations, I need the exact contour plot image. **Given the questions, here's how I would approach each step if I had the contour plot:** --- ### **Step-by-step approach (assuming typical contour behaviors)** --- ### **a) At the point (2, 1), is \( f_x(2, 1) \) positive or negative? Why?** **Method:** - **Partial derivative \( f_x(2, 1) \)** measures how \( f \) changes as \( x \) increases, keeping \( y \) constant. - **Observation:** - If moving along a small increment in \( x \) (say, from \( x=2 \) to \( x=2 + \delta \)), **the function value increases**, then \( f_x > 0 \). - Conversely, if it **decreases**, then \( f_x < 0 \). **How to determine from the contour plot:** - Find the point (2,1) on the plot. - Move slightly to the right (increase \( x \) slightly, e.g., to \( x=2.1 \)) at the same \( y=1 \). - Check the contour labels: - If the contour values **increase** from \( x=2 \) to \( x=2.1 \), then \( f_x > 0 \). - If they **decrease**, then \( f_x < 0 \). **Assumption (common case):** - Contours typically increase to the right or decrease to the right. - Suppose the contours **increase** as \( x \) increases at (2,1), then: **Conclusion:** \[ \boxed{ f_x(2,1) > 0 } \] --- ### **b) At the point (2, 1), is \( f_y(2, 1) \) positive or negative? Why?** **Method:** - \( f_y \) measures change in \( f \) as \( y \) increases. **How to determine:** - Move slightly upward in \( y \) (e.g., from \( y=1 \) to \( y=1.1 \)), at the same \( x=2 \). - Check the contour values: - If the values **increase** as \( y \) increases, then \( f_y > 0 \). - If they **decrease**, then \( f_y < 0 \). **Assumption:** - Suppose at \( y=1 \), the contour value is 5. - At \( y=1.1 \), the contour value is 6. - Since the value increases, then: \[ \boxed{ f_y(2,1) > 0 } \] --- ### **c) At the point (1, 2), what can you say about the partial derivatives \( f_x \) and \( f_y \)?** **Analysis:** - Similar steps: - To find \( f_x \) at (1,2), look at how \( f \) changes as you move in \( x \)-direction. - To find \( f_y \) at (1,2), look at how \( f \) changes as you move in \( y \)-direction. **Typical scenarios:** - If at (1,2), the contours are **closely spaced and increasing to the right**, \( f_x > 0 \). - If they **decrease to the right**, \( f_x < 0 \). - Similarly for \( y \). **Example:** - Suppose at (1,2), moving to \( x=1.1 \) increases the function value, then \( f_x(1,2) > 0 \). - Moving to \( y=2.1 \), the function decreases, then \( f_y(1,2) < 0 \). --- ### **d) Locate one point \( (x, y) \) where \( f_x(x, y) = 0 \). How do you know?** **Method:** - \( f_x = 0 \) when the function does **not change** as we move in the \( x \)-direction. - On the contour plot: - This occurs where the contours are **vertical lines** (Contour lines run parallel to the \( y \)-axis), meaning that moving horizontally doesn’t change \( f \). **How to find:** - Identify a vertical segment where the contour values are **constant** as \( x \) varies. - Pick a point on such a contour line. **Conclusion:** - For example, at \( y=1.5 \), if the contours are vertical lines at \( x=2.5 \) with the same value, then: \[ \boxed{ f_x(2.5, 1.5) = 0 } \] --- ### **e) Locate one point \( (x, y) \) where \( f_x(x, y) < 0 \). How do you know?** **Method:** - \( f_x < 0 \) when the function **decreases** as \( x \) increases. - On the contour plot: - The contours **decrease in value** as you move from left to right. **How to find:** - Find a region where moving right **reduces** the contour value. - For example, at \( y=1 \), moving from \( x=2 \) to \( x=2.1 \), functions decrease in value indicating \( f_x < 0 \). --- ## **Summary of Steps:** | Question | How to determine from the contour plot | Assumed conclusion based on typical contour behavior | |------------|-----------------------------------------|-----------------------------------------------------| | (a) \( f_x(2,1) \) | Increase in contours to the right | \( > 0 \) | | (b) \( f_y(2,1) \) | Increase in contours upward | \( > 0 \) | | (c) at (1,2) | Observe contour behavior in \( x \) and \( y \) directions | Depends on the slope of contours at that point | | (d) \( f_x=0 \) | Vertical contour lines | Point on a vertical contour line | | (e) \( f_x<0 \) | Contours decreasing to the right | Point where contours decrease in value moving right | --- ## **Important Note:** To provide **exact numerical answers** and **specific points**, I need the actual contour plot image. --- ### **Next step:** - **Upload the contour plot image**. - I will analyze it carefully and give precise answers with calculations. --- **Please upload the image or provide more details so I can assist further!**