Question

Determine whether $f\left(x\right)=-2x^2+12x+3$has a maximum or minimum value. Then find the maximum or minimum value.

Short answer

The quadratic function has a maximum value. The maximum value is$21$.

Step-by-step solution

Step 1. Given information. 

Determine whether $f\left(x\right)=-2x^2+12x+3$ has a maximum or minimum value. Then find the maximum or minimum value.

Step 2. Compare the function.

Compare

$$\begin{gathered} f\left(x\right)=-2x^2+12x+3\mathrm{to}f\left(x\right)=a{x}^2+bx+c \\ \mathrm{Then}\mathrm{a}=-2,\mathrm{b}=12\mathrm{and}\mathrm{c}=3 \end{gathered}$$

Since $a<0$the graph of f opens down, so the vertex is a maximum value occurs at

$x=-\frac{b}{2a}=-\frac{12}{2\left(-2\right)}=\frac{12}{4}=3$

The maximum value is

$f\left(-\frac{b}{2a}\right)=f\left(3\right)=-2{\left(3\right)}^2+12\left(3\right)+3=-18+36+3=21$

Step 3. Conclusion.  

The quadratic function has a maximum value. The maximum value is$21$.