Question

Determine whether each function has a maximum or minimum value. Then find the maximum or minimum value of the function.

$f\left(x\right)=-x^2+7$

Short answer

The maximum value of function is 7.

Step-by-step solution

Step 1. Use the concept.

Consider the function $f\left(x\right)=a{x}^2+bx+c,a\ne 0$, the x-coordinate of vertex is $\frac{-b}{2a}$.

The graph of $f\left(x\right)=a{x}^2+bx+c,a\ne 0$

• opens up and has a minimum value when $a>0$, and
• opens down and has a maximum value when $a<0$

Step 2. Given Information.

The given function is $f\left(x\right)=-x^2+7$

Step 3. Solution.

In the function $f\left(x\right)=-x^2+7$, we have $a=-1,b=0,c=7$

Here, $a=-1<0$

So, the graph opens down and has a maximum value.

The maximum value of the function is the y-coordinate of the vertex.

Thex-coordinate of the vertex is

$\begin{array}{l}\frac{-b}{2a}\\ =\frac{-0}{2(-1)}\mathrm{..........}\left[a=-1,b=0\right]\\ =0\end{array}$

Find they-coordinate of the vertex by evaluating the function for $x=0$.

$f\left(0\right)=-{\left(0\right)}^2+7=7$

So, the maximum value of function is 7.