Number of real roots of $3^{x^{22}}+28^{(1-x)^2} = 2020$, but without use of derivatives.

Question

How many real roots for $3^{x^{22}}+28^{(1-x)^2} = 2020$? Without derivatives.

I try study monotony of left function, but don't know where increasing and where decreasing.

Help me.

Answer 1

Let's define two functions $f,\ g : \mathbb{R}\to \mathbb{R}$ with $f(x) = 3^{x^{22}}$ and $g(x) = 2020 - 28^{(x-1)^2}$. It's easy to see the monotony of these functions:

• $f$ is decreasing over $(-\infty, 0]$ and increasing over $[0, \infty)$.
• $g$ is increasing over $(-\infty, 1]$ and decreasing over $[1, \infty)$.

Also, you have to notice that when $x \in [0,1]$, we have $f(x) \in [1,3]$ and $g(x) \in [1992, 2019]$, so there are no real roots in $[0,1]$.

That means there is at most a root in each of the intervals $(-\infty, 0]$ and $[1,\infty)$ respectively (because of the opposite monotony in these ranges).

Since

$$f(-1) = 3 > 0 > 2020-28^4 = g(-1)$$

and

$$f(0)=1<1992=g(0),$$

there is a root in $(-1,0)$. Also, because:

$$f(1)=3 < 2019=g(1)$$

and

$$f(2)=3^{2^{22}} > 1992 = g(2)$$

there exists one root in $(1,2)$. The final answer is exactly two roots.