Question

Hi, good evening Dear!! Can you help me to determine whether each function has a maximum or a minimum value, and find the maximum or minimum value of each function. Then state the domain and range of the function. Please.

Answer 1

`f(x)=-2x^2+4x-1`

To find a maximum or minimum, we have to derivate our function and set it equal to 0

`f^(\prime)(x)=-2x+4`
`\begin{gathered} -2x+4=0 \\ -2x=-4 \\ x=(-4)/(-2) \\ x=2 \end{gathered}`

Now that we have the point at x, we plug it into the initial function f(x) to find the point at y

`\begin{gathered} f(x)=y \\ f(2)=-(2)^2+4(2)-1 \\ y=-4+8-1 \\ y=3 \end{gathered}`

Now we know that our maximum or minimun point is (2,3)

To know if it is a maximum or a minimum we must perform the second derivative. Remember that if the sign of the second derivative is negative, our point is a maximum and if it is positive, our point is a minimum.

`f^(\prime)^(\prime)(x)=-2`

Now we know that our point is a maximum since its second derivative was negative. This can be seen in the graph, where our function belongs to a parabola that opens downwards

For the domain and range we have:

• The ,domain, of the function that is equivalent to all the values that x can take are, all the real ones

,

• The ,range, of the function that is equivalent to all the values it can take on and is defined from its maximum point to negative infinity, that is ,[3,-∞)