Find a polynomial of degree satisfying the following conditions:

Since must be a polynomial of degree we may write

where any of the may be 0 (since we could have a polynomial of degree strictly less than 5). First, let’s apply the condition to obtain

Now, let’s take the first two derivatives since we have conditions on and .

We can then apply the conditions and to obtain

So now we have and and so

Now we need to use the other three conditions

(If you know some linear algebra feel free to solve this in a more efficient way.) From the first equation we have

Plugging this into the second equation we have

Now plugging in our expressions for and into the third equation we have

Then using our expressions for and we have

Now, we have computed all of the constants so we can write down the formula for the polynomial