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Optimization
· Examples✏️ Worked Examples — Step by Step
E1
EasyMinimum of f(x) = x² + 4:
💡 x² ≥ 0; minimum when x = 0.
👁 Show Solution
x² ≥ 0, so min at x = 0 gives f = 0 + 4 = 4. ✅ Answer: 4 (at x = 0)
✅4
E2
MediumFind x that minimises f(x) = (x − 5)²
💡 Vertex of parabola.
👁 Show Solution
(x − 5)² ≥ 0; minimum 0 when x = 5. ✅ Answer: x = 5
✅5
E3
ChallengingGradient descent on f(x) = x² with learning rate 0.1, start x = 5. Value after 1 step?
💡 x ← x − α·∇f. ∇f = 2x.
👁 Show Solution
Step: x ← 5 − 0.1 × (2 × 5) = 5 − 1 = 4. ✅ Answer: 4
✅4