Question

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

$f\left(x\right)=x-2x^2-1$

Short answer

The function has maximum value that is -0.875.

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 $f\left(x\right)=x-2x^2-1=-2x^2+x-1$

Step 3. Solution.

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

Here, $a=-2<0$

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

$\begin{array}{l}\frac{-b}{2a}\\ =\frac{-\left(1\right)}{2(-2)}\mathrm{..........}\left[a=-2,b=1\right]\\ =0.25\end{array}$

Find they-coordinate of the vertex by evaluating the function for $x=0.25$.

$f(0.25)=-2{\left(0.25\right)}^2+\left(0.25\right)-1=-0.875$

So, the maximum value of function is -0.875.