Mathematico | Mathematico

The curve

y = \frac{1}{4^{x}} - 2^{2 - x} + 3

is sketched below:

Curve with minimum point

Find the coordinates of the minimum point of the curve.

The first term can be written as

\begin{aligned} \frac{1}{4^{x}} & = \frac{1}{(2^{x})^{2}} \\ = {(\frac{1}{2^{x}})}^{2} \end{aligned}

Can you write the second term, $- 2^{2 - x}$ , in terms of $\frac{1}{2^{x}}$ as well?

We have

\begin{aligned} - 2^{2 - x} & = - 2^{2} 2^{- x} \\ = - 4 (\frac{1}{2^{x}}) \end{aligned}

With hint (a) in mind also, we can write the equation of the curve as

y = {(\frac{1}{2^{x}})}^{2} - 4 (\frac{1}{2^{x}}) + 3

This is a quadratic in disguise. Try letting $u = \frac{1}{2^{x}}$ and complete the square to find the turning point.