Math Problem Statement
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Solution
To solve the given problem, we need to:
-
Find the critical points of the function .
- This involves computing the first partial derivatives and , setting them to zero, and solving for .
-
Classify the critical points using the second derivative test, which involves:
- Computing the second partial derivatives , , and .
- Using the determinant of the Hessian matrix: .
- If and , the point is a local minimum.
- If and , the point is a local maximum.
- If , the point is a saddle point.
- If , the test is inconclusive.
Let's start by computing the first partial derivatives and finding the critical points. sympy import symbols, diff, solve, hessian, det
Define variables and function
x, y = symbols('x y') f = x2 * y + 3 * y2 - 2 * x * y + 2
Compute first partial derivatives
fx = diff(f, x) fy = diff(f, y)
Solve for critical points (fx = 0 and fy = 0)
critical_points = solve((fx, fy), (x, y)) critical_pointsThe critical points of the function are:
Next, let's classify these points using the second derivative test by calculating the Hessian matrix and its determinant . Compute second partial derivatives fxx = diff(fx, x) fyy = diff(fy, y) fxy = diff(fx, y)
Hessian determinant D = fxx * fyy - (fxy)^2
D = fxx * fyy - fxy**2
Evaluate D and fxx at each critical point to classify them
classification = [] for point in critical_points: D_val = D.subs({x: point[0], y: point[1]}) fxx_val = fxx.subs({x: point[0], y: point[1]})
if D_val > 0:
if fxx_val > 0:
classification.append((point, 'Local Minimum'))
elif fxx_val < 0:
classification.append((point, 'Local Maximum'))
elif D_val < 0:
classification.append((point, 'Saddle Point'))
else:
classification.append((point, 'Inconclusive'))
classificationThe classification of the critical points is as follows:
- : Saddle Point
- : Local Minimum
- : Saddle Point
Thus, the answers are:
- Local Maximum: DNE
- Local Minimum:
- Saddle Point:
Would you like to explore the steps in more detail or have any questions?
Here are 5 related questions to expand your understanding:
- How do you compute the Hessian matrix in multivariable calculus?
- Why does the sign of the determinant classify critical points?
- What happens when in the second derivative test?
- How can a function have no local maxima but still have other critical points?
- What are practical applications of finding and classifying critical points?
Tip: Always check your second derivative test calculations carefully to avoid classification errors.
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Critical Points
Second Derivative Test
Formulas
f_x = 0 and f_y = 0 to find critical points
D = f_xx f_yy - (f_xy)^2 for Hessian determinant
Theorems
Second Derivative Test for classifying critical points
Suitable Grade Level
Undergraduate
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