## Task 2 - 201232

Let $f$ be the function defined by

$$
f(x)=x^{4}-(x+1)^{4}-(x+2)^{4}+(x+3)^{4}
$$

where the domain of $f$ is

a) the set of all integers,

b) the set of all real numbers.

Investigate, for both case a) and case b), whether the function $f$ attains a minimum value, and determine, if applicable, this minimum value in each case.
From the representation

$$
f(x)=\left(\left(x+\frac{3}{2}\right)-\frac{3}{2}\right)^{4}-\left(\left(x+\frac{3}{2}\right)-\frac{1}{2}\right)^{4}-\left(\left(x+\frac{3}{2}\right)+\frac{1}{2}\right)^{4}+\left(\left(x+\frac{3}{2}\right)+\frac{3}{2}\right)^{4}
$$

we see that $f$ is axis-symmetric with respect to $x=-\frac{3}{2}$ and is at most a quadratic polynomial. In particular, $f$ has its global minimum or maximum at $x=-\frac{3}{2}$.

From $f(-1)=f(-2)=16$ and $f\left(-\frac{3}{2}\right)=10$, it follows that $f$ is a parabola opening upwards and thus has its real minimum at $x=-\frac{3}{2}$ and its minimum over the integers at $x=-1, x=-2$.

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