+870 Let f be the function defined on [0,1] by \(f:x \mapsto \sqrt{x}-x\)
Determinate the maximum of function f, and justify rigorously your answer.
Ye'll have to follow the reduction to the canonical form of a quadratic polynomial.
HINT:
\(\forall x \in [0,1], \\\sqrt{x} \geq x\)
Different forms of a quadratic polynomial:
Expanded form: ax²+bx+c (a,b,c being real numbers)
Canonical form: a(x-α)²+β
\(\alpha = \frac{-b}{2a} \\\beta = \frac{-\Delta}{4a} \\\Delta=b^2-4ac\)
Factored form: a(x-x1)(x-x2) if discriminant Δ>0
a(x-x0)² if Δ=0
The factored form exists only for positive discriminants.
If a>0 then α is the minimum of the function
If a<0 then α is the maximum of the function
