This maths challenge comes from the University of Cambridge. It’s a test for students looking to study computer science and maths-related subjects at the prestigious institution.

Specifically, we are tasked with finding the minimum value of a function.

There are 6 choices but only 1 is the correct answer.

Once again, this is a good moment to pause the article and give the problem a go yourself. When you’re ready, keep reading for the solution. And if you come up with your own approach, feel free to share it in the comments — I’d love to see how you tackled it.

Don’t forget to subscribe to our YouTube channel for more maths puzzles, it’s my goal to reach 100k subscribers someday. 

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Solution

Let’s solve this together.

We want to find the minimum value of

Whenever you see different powers of the same base, ask yourself:

Can I replace the repeated expression with a single variable?

Here, both terms involve $2^x$, so let’s set

Since $2^x$ is always positive, we know

Now rewrite the expression.

So our function becomes
$$$$

Much nicer!

Now comes the question:

How do we find the minimum of a quadratic?

The quickest method is completing the square.

Now notice something important.

The square term,

can never be negative. Its smallest possible value is $0$0.

So the entire expression is smallest when

which happens when

Does this value actually occur?

Since

we have

so

Substituting back,

Therefore, the minimum value is

And that’s our answer. How amazing. Hey, don’t forget to clap👏 the article as a token of appreciation. Thank you 🦁