Question

1. Does the function have a minimum or maximum value?

Answer 1

the tale-tell fellow is the coefficient of the squared variable, namely -2 of x². Since the coefficient is negative, the parabola opens downwards, like the one in the picture below, so it goes up up up, reaches a maximum, then goes back down down down.

$\bf \textit{vertex of a vertical parabola, using coefficients}\\\\g(x)=\stackrel{\stackrel{a}{\downarrow }}{-2}x^2\stackrel{\stackrel{b}{\downarrow }}{+20}x\stackrel{\stackrel{c}{\downarrow }}{-49}\qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)\\\\\\\left(-\cfrac{20}{2(-2)}~~,~~-49-\cfrac{20^2}{4(-2)} \right)\implies \left(5~~,~~-49+\cfrac{400}{8} \right)\\\\\\(5~~,~~-49+50)\implies (5,1)$