College Algebra, Chapter 4, 4.1, Section 4.1, Problem 50

A quadratic function $f(x) = 1 + x - \sqrt{2} x^2$.

a.) Find the maximum or minimum value of the quadratic function $f$, using a graphing device.







Based from the graph, the function has a maximum value of $1.2$.

b.) Find the exact maximum or minimum value of $f$.

Using the quadratic equation $ax^2 + bx + c$ with $a = - \sqrt{2}$ and $b = 1$. Thus, the maximum or minimum value occurs at

$\displaystyle x = - \frac{b}{2a} = - \frac{1}{2(- \sqrt{2})} = \frac{1}{2 \sqrt{2}}$

Since $a < 0$, the function has a maximum value

$\displaystyle f \left( \frac{1}{2 \sqrt{2}} \right) = 1 + \frac{1}{2 \sqrt{2}} - \sqrt{2} \left( \frac{1}{2 \sqrt{2}} \right)^2 = 8 + \frac{\sqrt{2}}{8} = 1.18$