Question

ind the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola.

y = 7 − x²

Answer 1

$S(t)=2\cdot t\cdot \ (7-t^2) =\space area\space of\space rectangle\\ S’(t)=2\cdot (7-t^2) -4\cdot t^2=14-6\cdot t^2\space is\space the \space derivative\\ S’(t)=14-6\cdot t^2=0 \space \\ we \space find\space zeros\space of\space the \space derivative\\ t0=\sqrt\frac{7}{3} -\space unique \space positive \space root\\ S''(t0)=-12\cdot t0=-12\sqrt\frac{7}{3}<0, \space the \space second \space derivative\\ is\space negative,\space therefore\space t0\space is \space maximal\space point\\ Base\space of\space the \space largest\space rectangle \space is\space a0=2\cdot\sqrt\frac{7}{3}\\ The\space height\space is \space h0=7-\frac{7}{3}=\frac{14}{3}$

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