For each of the functions below, verify whether they are (1) convex, (2) strictly convex, (3) concave or not: A. f(x) =

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For Each Of The Functions Below Verify Whether They Are 1 Convex 2 Strictly Convex 3 Concave Or Not A F X 1 (111.18 KiB) Viewed 25 times

For each of the functions below, verify whether they are (1) convex, (2) strictly convex, (3) concave or not: A. f(x) = -2, x ER B. f(x) = sin(x), x ER C. f(x) = ReLu(ax + b), x € R D. f(x) = ReLu(a₂x₂ (a₁x₁ + b₁) + b₂), x = R² E. f(x) = ex, x ER F. f(x) = exp(-a¹x) + || Ax − b||², x € R² G. f(x) = x¹ Ax, x E R² where A := ( ), ReLu(x):= 0, if x < 0; x, otherwise. a, b, a, b, ER, a, b E R². Answer the following questions by dragging and dropping the correct options. 1. Given the function A. above, which are all of its properties? 2. Given the function B. above, which are all of its properties? 3. Given the function C. above, which are all of its properties? 4. Given the function D. above, which are all of its properties? 5. Given the function E. above, which are all of its properties? 6. Given the function F. above, which are all of its properties? 7. Given the function G. above, which are all of its properties? 8. We consider C₁ and C₂ two convex sets in Rd. We define C₁ + C₂ := {x₁ + x₂, X₁ € C₁, X₂ € C₂}. Is C₁ + C₂ a convex set? 9. HP(a, b) = {x € Rª | a¹x = b} is a convex set. Is this true? (a) convex (b) convex + strictly convex (c) concave (d) convex + concave (e) none of these properties True False