How can I show that $\max_{t \in [0,1]}|t^{\alpha} - c| = \max(c,1-c)$ for $t \in [0,1]$?

Question

Let $t \in [0,1]$ and $\alpha >0$. In my notes, our professor wrote:

$$\max_{t \in [0,1]}|t^{\alpha} - c| = \max(c,1-c)$$ when $c$ is a constant in $[0,1]$. How can I show this?

Answer 1

Think about the general shape of $t^\alpha$ as a diagram. As $\alpha >0$ you will notice that this is a strictly non-decreasing function on the interval $[0,1]$, so would be $t^\alpha - c$.

Now (and you maybe need to think about this, maybe draw it out to convince yourself) the maximum value of $|t^\alpha-c|$ takes value at either $t=0 $ or $t=1$ due to this monotonacity. The rest should be rather clear to you now.