Question

Over what domain interval is the graph of f(x)=(x-3)^2 increasing?

Answer 1

The increasing interval of a function can be founded by taking the derivative of the function and then calculating where the derivative is possitive.

We have the function:

`f(x)=(x-3)^2`

The derivative, by the chain rule is:

`f^(\prime)(x)=2(x-3)`

Now we need to find the interval where the derivative is possitive. Let's find the root:

`\begin{gathered} 0=2(x-3) \\ 0=x-3 \\ x=3 \end{gathered}`

The derivative is 0 when x = 3. Now lets evaluate the derivative in a number greater than 3, if it's possitive, the increasing interval will be (3,∞) If it's negative, the interval will be (-∞, 3)

Let's evaluate for x = 4:

`f^(\prime)(4)=2(4-3)=2`

Then, the increasing interval is (3,∞)