O是直角坐标原点,A,B是抛物线y2＝2x上异于顶点的两动点,且OA⊥OB.点A,B在什么位置时,△AOB的面积最小

设A(a2/2,a),B(b2/2,b)
向量OA = (a²/2,a),向量OB(b²/2,b)
OA⊥OB,向量OA•向量OB = (a²/2)(b²/2) + ab = ab(ab/4 + 1) = 0
ab≠0,ab/4 + 1 = 0,ab = -4
△AOB的面积S = (1/2)|OA|*|OB| = (1/2)√[(a²/2)² + a]*√[(b²/2)² + b²]
= (1/2)√[a⁴/4 + a²)(b⁴/4 + b²)]
= (1/2)√(a⁴b⁴/16 + a⁴b²/4 + a²b⁴/4 + a²b²)
= (1/2)√[16 + a²b²(a² + b²)/4 + 16]
= (1/2)√(32 + 4(a² + b²)]
= √(8 + a² + b²)
= √(8 + a² + 16/a²)
a² = 16/a² (即a = ±2)时,S最小
只须取a = 2,因为两个解关于x轴对称,AB⊥x轴