Step by step
The question
Question: Two numbers add up to 30.
Find the largest product.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Name the unknown
Call the first number x.
They add up to 30, so the second number is 30 − x.
2 Write the product
The product is the first number times the second.
The x in front of the bracket multiplies every part inside.
Now we have P(x) = 30x − x².
3 Solve f′(x) = 0
The biggest product is at the top of the curve.
There it stops going up and starts going down, so for a moment it is flat: the slope is 0.
The power rule gives P′(x) = 30 − 2x.
Set it equal to 0.
Add 2x to both sides, so they stay equal.
To remove times 2, we divide by 2.
30 divided by 2 is 15.
4 Is it the biggest?
Before 15 the slope is positive: going up.
After it, negative: going down.
Up, then down: a maximum.
Careful!
x is only the first number.
The question asks for the product.
5 Answer the question
The other number is 15.
So the largest product is 225.
6 Check
Let's check with x a bit smaller: 14 gives 224.
And x = 16 gives 224.
Both are smaller than 225.
Right!
7 Pick the choice
Our answer is largest product = 225.
That is choice A.
8 Why not the others
B is 15.
Wrong: that is each number, not the product.
C is 900.
Wrong: 30 × 30: the two numbers add up to 30, so each is 15.
D is 200.
Wrong: used 10 and 20: 15 and 15 give more.
Let's look back.
First: name the unknown.
Then: write the product.
Then: solve f′(x) = 0.
Then: is it the biggest?
Then: answer the question.
Last: check.