(20 points) Show that the stationary point (zero gradient) of the func- tion f(x, x) =...

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(20 points) Show that the stationary point (zero gradient) of the func- tion f(x₁, x₂) = 2x² - 4x1x2 +1.5x² + x2 is a saddle (with indefinite Hessian). Find the directions of downslopes away from the saddle. To do this, use Taylor's expansion at the saddle point to show that f(x₁, x₂) = f(1, 1) + (adx₁ − bəx₂) (cdx₁ - dəx₂), - with some constants a, b, c, d and Oxi = Xi 1 for i=1, 2. Then the directions of downslopes are such (dx₁, x₂) that ƒ(x1, x2) — ƒ(1, 1) = (aðx₁ — bəx2)(cəx1 — dəx2) < 0. (20 points) Show that the stationary point (zero gradient) of the func- tion f(x₁, x₂) = 2x² - 4x1x2 +1.5x² + x2 is a saddle (with indefinite Hessian). Find the directions of downslopes away from the saddle. To do this, use Taylor's expansion at the saddle point to show that f(x₁, x₂) = f(1, 1) + (adx₁ − bəx₂) (cdx₁ - dəx₂), - with some constants a, b, c, d and Oxi = Xi 1 for i=1, 2. Then the directions of downslopes are such (dx₁, x₂) that ƒ(x1, x2) — ƒ(1, 1) = (aðx₁ — bəx2)(cəx1 — dəx2) < 0.

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