Step by step
The question
Question: Two positive numbers multiply to 64.
Find the smallest possible sum.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Draw and name
Call the first number x, and the second y.
2 Write the sum
The sum is x plus y.
3 Use only one letter
They multiply to 64: x times y is 64.
So y = 64 over x.
Put it in place of y.
4 Which x make sense
The numbers are positive: x is more than 0.
5 Slope = 0
The smallest value is where the slope is 0.
64/x is 64x to the power −1.
The power rule gives −64/x².
And x gives 1.
Set it to 0.
Add 64/x² to both sides.
To clear the fraction, times both sides by x², so they stay equal.
x squared is 64, so x is 8.
Minus 8 squared is also 64, but x must be positive.
Careful!
The numbers must be positive.
Only x = 8 is allowed.
The right way: x = 8 (not −8).
6 Make sure it is the smallest
Before 8, the slope is negative: going down.
After it, positive: going up.
Down, then up: the bottom.
A minimum.
7 Answer the question
The other number: 64/8 is 8.
The same number!
So the smallest sum is 16.
8 Check
Let's check with x a bit smaller: 7 gives 16.14.
And x = 9 gives 16.11.
Both are bigger than 16.
Right!
9 Pick the choice
Our answer is smallest sum = 16.
That is choice C.
10 Why not the others
A is 8.
Wrong: that is each number, not the sum.
B is 65.
Wrong: used 1 and 64: that sum is the largest here.
D is 20.
Wrong: used 4 and 16: 8 + 8 is smaller.
Let's look back.
First: draw and name.
Then: write the sum.
Then: use only one letter.
Then: which x make sense.
Then: slope = 0.
Then: make sure it is the smallest.
Then: answer the question.
Last: check.