Question Number 103828 by ~blr237~ last updated on 17/Jul/20

     min{ ∫_0 ^1 (x^3 −px−q)^2 dx ,  (p,q)∈R^2  }

Answered by bobhans last updated on 17/Jul/20

(x^3 −(px+q))^2 = x^6 −2x^3 (px+q)+(px+q)^2   = x^6 −2px^4 −2qx^3 +p^2 x^2 +2pqx+q^2   ∫_0 ^1 (x^6 −2px^4 −2qx^3 +p^2 x^2 +2pqx+q^2 ) dx =  (1/7)−((2p)/5)−(q/2)+(p^2 /3)+pq+q^2  = f(p,q)  (∂f/∂p) = −(2/5)+((2p)/3)+q = 0 ⇒q=(2/5)−((2p)/3)  (∂f/∂q) = −(1/2)+p+2q=0⇒−(1/2)+p+(4/5)−((4p)/3)=0  (p/3) = (3/(10)) ⇒p = (9/(10)) ∧q = (2/5)−(2/3).(9/(10))  q = −(1/5).