# Find the values of b such that the function has the given maximum or minimum val

Find the values of b such that the function has the given maximum or minimum value.
$$\displaystyle{f{{\left({x}\right)}}}=-{x}^{{{2}}}+{b}{x}-{75}$$; Maximum value: 25

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Macsen Nixon
The maximum or minimum of a quadratic function $$\displaystyle{y}={a}{x}^{{{2}}}+{b}{x}+{c}$$ occurs at the vertex which has an x-coordinate given by:
$$\displaystyle{x}=-{\frac{{{b}}}{{{2}{a}}}}$$
From the given, a=-1 and b=b sp we have:
$$\displaystyle{x}=-{\frac{{{b}}}{{{2}{\left(-{1}\right)}}}}={\frac{{{b}}}{{{2}}}}$$
Since a=-1
$$\displaystyle-{\left({\frac{{{b}}}{{{2}}}}\right)}^{{{2}}}+{b}{\left({\frac{{{b}}}{{{2}}}}\right)}-{75}={25}$$
Solve for b:
$$\displaystyle-{\frac{{{b}^{{{2}}}}}{{{4}}}}+{\frac{{{b}^{{{2}}}}}{{{2}}}}-{75}={25}$$
$$\displaystyle{\frac{{{b}^{{{2}}}}}{{{4}}}}={100}$$
$$\displaystyle{b}^{{{2}}}={400}$$
$$\displaystyle{b}=\pm\sqrt{{{400}}}$$
$$\displaystyle{b}=\pm{20}$$
Result:
$$\displaystyle{b}=\pm{20}$$