Find all critical points for f(x,y) = (x^4) + (y^4) - (4x) - (32y) + 10 and clas

Question

Find all critical points for f(x,y) = (x^4) + (y^4) - (4x) - (32y) + 10 and classify as max, min, or saddle points.

Explanation / Answer

1) Critical points insider the region: Set the first partial derivatives of f equal to 0: f_x = 4x - 4, f_y = 2y - 4 Setting these to 0 yields (x, y) = (1, 2). Note that this is on the boundary line y = 2; so we'll ignore it (see below). ---------------------- 2) Critical points on boundary lines. i) x = 0 for y in [0, 2]: ==> g(y) = f(0, y) = y^2 - 4y + 1 g'(y) = 2y - 4 = 0 ==> y = 2 So, the critical point is (x, y) = (0, 2). ---- ii) y = 2 for x in [0, 1]: ==> g(x) = f(x, 2) = 2x^2 - 4x - 3 g'(x) = 4x - 4 = 0 ==> x = 1 So, the critical point is (x, y) = (1, 2). ---- iii) y = 2x for x in [0, 1]: ==> g(x) = f(x, 2x) = 6x^2 - 12x + 1 g'(x) = 12x - 12 ==> x = 1 So, the critical point is (x, y) = (1, 2). ---- 3) The corner points of the region are (x, y) = (0, 0), (1, 2), (0, 2). (Note that the critical points from 2 are also corner points!) ---- 4) Testing all of the points for extrema: f(0, 0) = 1

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.