Question

Let \( f, g: \mathbb{R}^2 \to \mathbb{R} \) be defined by
\[ f(x, y) = x^2 - \frac{3}{2}xy^2 \quad \text{and} \quad g(x, y) = 4x^4 - 5x^2y + y^2 \] for all \( (x, y) \in \mathbb{R}^2 \).
Consider the following statements:
P: \( f \) has a saddle point at (0, 0).
Q: \( g \) has a saddle point at (0, 0).
Then

• A. both P and Q are TRUE
• B. P is FALSE but Q is TRUE
• C. P is TRUE but Q is FALSE
• D. both P and Q are FALSE

Hint:

When the second derivative test for multivariable functions yields a discriminant D=0, don't assume anything. Test the function's behavior along simple paths like \(y=0\), \(x=0\), \(y=x\), or \(y=mx^k\). If you find paths where the function has opposite behaviors (max vs. min), you've proven it's a saddle point.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To determine if a critical point is a local maximum, local minimum, or saddle point, we use the second partial derivative test. If the test is inconclusive (Discriminant D = 0), we must analyze the function's behavior along different paths approaching the critical point. A point is a saddle point if the function has a local maximum along one path and a local minimum along another path.
Step 2: Key Formula or Approach:
For a function \(h(x, y)\) with a critical point at \((a, b)\): 1. Find the first partial derivatives \(h_x\) and \(h_y\) and verify they are zero at \((a, b)\). 2. Calculate the second partial derivatives \(h_{xx}\), \(h_{yy}\), and \(h_{xy}\). 3. Compute the discriminant \(D(x, y) = h_{xx}h_{yy} - (h_{xy})^2\). 4. If \(D(a, b)<0\), it's a saddle point. If \(D(a, b) = 0\), the test is inconclusive.
Step 3: Detailed Calculation:
Analysis of Statement P for f(x, y) = \(x^2 - \frac{3}{2}xy^2\):
First, find the critical points.
\(f_x = 2x - \frac{3}{2}y^2\). At (0,0), \(f_x = 0\).
\(f_y = -3xy\). At (0,0), \(f_y = 0\).
So, (0,0) is a critical point.
Now, find the second partial derivatives.
\(f_{xx} = 2\), \(f_{yy} = -3x\), \(f_{xy} = -3y\).
At (0,0): \(f_{xx}(0,0) = 2\), \(f_{yy}(0,0) = 0\), \(f_{xy}(0,0) = 0\).
The discriminant \(D(0,0) = f_{xx}f_{yy} - (f_{xy})^2 = (2)(0) - (0)^2 = 0\).
The test is inconclusive. We test paths near (0,0), where \(f(0,0)=0\).
- Along the x-axis (\(y=0\)): \(f(x, 0) = x^2\). This is always \(\ge 0\), indicating a local minimum.
- Along the parabola \(x = y^2\): \(f(y^2, y) = (y^2)^2 - \frac{3}{2}(y^2)y^2 = y^4 - \frac{3}{2}y^4 = -\frac{1}{2}y^4\). This is always \(\le 0\), indicating a local maximum.
Since \(f(x,y)\) increases along one path and decreases along another, (0,0) is a saddle point for \(f\). Thus, P is TRUE.
Analysis of Statement Q for g(x, y) = \(4x^4 - 5x^2y + y^2\):
First, find the critical points.
\(g_x = 16x^3 - 10xy\). At (0,0), \(g_x = 0\).
\(g_y = -5x^2 + 2y\). At (0,0), \(g_y = 0\).
So, (0,0) is a critical point.
Now, find the second partial derivatives.
\(g_{xx} = 48x^2 - 10y\), \(g_{yy} = 2\), \(g_{xy} = -10x\).
At (0,0): \(g_{xx}(0,0) = 0\), \(g_{yy}(0,0) = 2\), \(g_{xy}(0,0) = 0\).
The discriminant \(D(0,0) = g_{xx}g_{yy} - (g_{xy})^2 = (0)(2) - (0)^2 = 0\).
The test is inconclusive. We analyze the function's form near (0,0), where \(g(0,0)=0\).
We can rewrite \(g(x,y)\) as \(g(x,y) = (y - \frac{5}{2}x^2)^2 - \frac{25}{4}x^4 + 4x^4 = (y - \frac{5}{2}x^2)^2 - \frac{9}{4}x^4\).
- Along the y-axis (\(x=0\)): \(g(0, y) = y^2\). This is always \(\ge 0\), indicating a local minimum.
- Along the parabola \(y = \frac{5}{2}x^2\): \(g(x, \frac{5}{2}x^2) = (0)^2 - \frac{9}{4}x^4 = -\frac{9}{4}x^4\). This is always \(\le 0\), indicating a local maximum.
Since \(g(x,y)\) increases along one path and decreases along another, (0,0) is a saddle point for \(g\). Thus, Q is TRUE.
Step 4: Final Answer:
Both statements P and Q are TRUE.
Step 5: Why This is Correct:
For both functions, the second derivative test at (0,0) is inconclusive. By analyzing the behavior of each function along different paths through the origin, we found that in both cases, the function value could be greater or less than the value at the origin. This is the definition of a saddle point.