Question

Suprafata minima marginita de f si Ox intre x=0 si x=1 ???

Answer 1

$f(x) = a^2x^2+ax+1 \\ \\ f'(x) = 2a^2x+a \\ f'(x) = 0 \Rightarrow 2a^2x+a = 0 \Rightarrow 2a^2x = -a \Rightarrow x =-\dfrac{1}{2a} \\ f(-\dfrac{1}{2}a) = a^2\cdot \dfrac{1}{4a^2}-\dfrac{1}{2}+1 = \dfrac{3}{4} \Rightarrow f(x) \geq \dfrac{3}{4} \\ \\ \int_{0}^1(a^2x^2+ax+1)\,dx = \Big(\dfrac{a^2x^3}{3}+\dfrac{ax^2}{2}+x\Big)\Big|_{0}^1 = \dfrac{a^2}{3}+\dfrac{a}{2}+1 \\ \\= \dfrac{1}{6}\cdot (2a^2+3a+6)\\ \\ g(a) = 2a^2+3a+6 \\ g'(a) = 4a+3 \Rightarrow 4a+3 = 0 \Rightarrow a = -\dfrac{3}{4}$

$\Rightarrow g_{min} = g\Big(-\dfrac{3}{4}\Big) \\ \\ \Rightarrow (d)~a = - \dfrac{3}{4}$