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Question

# A certain viral infection causes a fever that typically lasts 6 days. A model of the fever $\left( \text{in}^{\circ} \mathrm{F}\right)$ on day $x, 1 \leq x \leq 6$ is$F(x)=-\frac{2}{3} x^{2}+\frac{14}{3} x+96$According to the model, on what day should the maximum fever occur? What is the maximum fever?

Solution

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The model of the viral infection fever ($^\circ F$) on day $x$ is given by, $F(x)=-\dfrac{2}{3}x^2+\dfrac{14}{3}x+96,\quad 1\le x\le 6.$

We see by the negative in the $x^2$-term that this defines a parabola opening downward, so the maximum revenue is at the vertex. The $x$-coordinate of the vertex is $x=-\dfrac{b}{2a}= -\dfrac{\dfrac{14}{3}}{2(-\dfrac{2}{3})}=\dfrac{7}{2}.$

Therefore, the maximum fever occur on the third day.

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