Find the x-coordinate minimizing f(x,y)=x²+2y²−6x.
Answer and explanation
3
The x derivative is 2x−6; setting it to zero gives 3, and the Hessian is positive definite.
Subjects
Optimization
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Sample exercises
For a differentiable convex function, a zero gradient implies what?
- A global minimum
- Always a strict maximum
- Always exactly two minima
- No information ever
Answer and explanation
A global minimum
The supporting-hyperplane inequality gives f(y)≥f(x).
Prove that a strictly convex function has at most one minimizer on a convex domain.
Answer and explanation
If two distinct minimizers existed, strict convexity at their midpoint would give a value strictly below their common minimum, a contradiction.
Credit the midpoint lying in the domain and strict inequality.
Minimize x²+y² subject to x+y=6. Find the minimum value.
Answer and explanation
18
The minimum occurs at x=y=3 by completing a square or Lagrange multipliers; value 9+9=18.
Is the feasible set x+y≤1 in R² convex?
- Only its boundary is convex
- Only for positive coordinates
- Yes
- No, because it is unbounded
Answer and explanation
Yes
A linear inequality is preserved under convex combinations.
Why must a proposed constrained optimum be checked for feasibility?
Answer and explanation
Stationarity equations alone can produce points outside the allowed set; an infeasible point cannot solve the constrained problem.
Credit distinction between algebraic candidates and admissible solutions.
Study notes
Convexity and constrained optimization
Values on a line segment lie below the corresponding chord of function values.
Advanced universityConvexity and constrained optimization · revision notes
Values on a line segment lie below the corresponding chord of function values.
Advanced universityStationary point of a positive-definite quadratic
This separable quadratic has positive curvature in each coordinate.
Advanced universityStationary point of a positive-definite quadratic · formula reference
This separable quadratic has positive curvature in each coordinate.
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