Question

Determine the second-order Taylor formula for the given function about the given point (x0, y0). f(x, y) = 9 x2 + y2 + 1 , where x0 = 0 and y0 = 0

O f(h1, h2) = 9 − 9 2 h12 − 9 2 h22 + R2(0, h)

O f(h1, h2) = 9 − 9h12 − 9h22 + R2(0, h)

O f(h1, h2) = 9 + 9h12 + 9h22 + R2(0, h)

O f(h1, h2) = 9 − 9h1 − 9h2 − 9h12 − 9h22 + R2(0, h)

O f(h1, h2) = 9 + 9 2 h12 + 9 2 h22 + R2(0, h)

Answer 1

The second-order Taylor formula for the given function f(x, y) = 9x² + y² + 1 about the point (x0, y0) is f(h1, h2) = 9 + 9h₁² + 9h₂² + R2(0, h).

Step-by-step explanation:

The second-order Taylor formula for the given function f(x, y) = 9x² + y² + 1 about the point (x0, y0) is given by:

f(h1, h2) = f(x0, y0) + f1(x0, y0)(h1-x0) + f2(x0, y0)(h2-y0) + (1/2)f11(x0, y0)(h1-x0)² + f12(x0, y0)(h1-x0)(h2-y0) + (1/2)f22(x0, y0)(h2-y0)² + R2(0, h)

In this case, since x0 = 0 and y0 = 0, the second-order Taylor formula becomes:

f(h1, h2) = 9 + 9h₁² + 9h₂² + R2(0, h)