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Local extrema and saddle points practice problems

Critical points of f(x, y), classified with the second derivative test. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.

Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 2 x - y^{2} + 4 y - 3 \).
Problem 10.39easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = - x^{2} - 2 x - 2 y^{2} - 4 y - 5 \).
Problem 10.40easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = - x^{2} + 4 x - 2 y^{2} + 12 y - 21 \).
Problem 10.41easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 4 x + y^{2} - 2 y + 4 \).
Problem 10.42easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} + 6 x - y^{2} + 9 \).
Problem 10.43easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} + 4 x - y^{2} + 6 y - 5 \).
Problem 10.44easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} + 6 x + y^{2} + 2 y + 10 \).
Problem 10.45easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = - x^{2} - 2 y^{2} + 12 y - 20 \).
Problem 10.46easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - y^{2} - 6 y - 9 \).
Problem 10.47easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = - x^{2} - 4 x - 2 y^{2} + 4 y - 5 \).
Problem 10.48easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 6 x + y^{2} - 6 y + 21 \).
Problem 10.49easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 6 x + y^{2} + 4 y + 15 \).
Problem 10.50easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = - x^{2} - 6 x - 2 y^{2} - 8 y - 19 \).
Problem 10.51easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} + 6 x + y^{2} - 6 y + 15 \).
Problem 10.52easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = - x^{2} - 2 x - 2 y^{2} + 8 y - 7 \).
Problem 10.53easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = - x^{2} - 2 y^{2} - 8 y - 8 \).
Problem 10.93easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 4 x + y^{2} - 2 y + 6 \).
Problem 10.94easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 6 x - y^{2} + 9 \).
Problem 10.95easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 4 x + y^{2} - 6 y + 13 \).
Problem 10.96easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} + 2 x - y^{2} + 2 y \).
Problem 10.97easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = - x^{2} - 2 y^{2} + 8 y - 6 \).
Problem 10.98easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 4 x + y^{2} + 2 y + 6 \).
Problem 10.99easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} + 4 x + y^{2} - 6 y + 16 \).
Problem 10.100easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = - x^{2} - 4 x - 2 y^{2} - 8 y - 13 \).
Problem 10.101easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} + 2 x - y^{2} - 4 y - 3 \).
Problem 10.102easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} + 4 x + y^{2} - 2 y + 2 \).
Problem 10.103easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 6 x + y^{2} + 6 y + 17 \).
Problem 10.104easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} + 6 x - y^{2} + 4 y + 5 \).
Problem 10.105easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 2 x - y^{2} - 6 y - 8 \).
Problem 10.106easy✓ Every equation proved
Find and classify the critical points of \( \displaystyle f(x, y) = x^{2} - 6 x - y^{2} + 2 y + 8 \).
Problem 10.107easy✓ Every equation proved