4. Given the function $f(x)=a+b c^{x}$. Determine the real numbers $a, b, c$, if $f(0)=5$, $f(1)=14$ and $f(2)=50$.
4. Insert the given data $f(0)=5, f(1)=14$ and $f(2)=50$ into the function $f(x)$. We get the system of equations $5=a+b, 14=a+b c, 50=a+b c^{2}$. Using the substitution method, we obtain the solutions for $c$: $c_{1}=4$ and $c_{2}=1$. The solution $c_{2}$ is discarded due to the undefined fraction $\frac{9}{c-1}$. We then calculate the remaining values $a=2, b=3$ and write the function as $f(x)=2+3 \cdot 4^{x}$.

Writing the system of equations from the given data $5=a+b, 14=a+b c, 50=a+b c^{2}$ ..... 1 point
Calculating the unknown $b=\frac{9}{c-1}=\frac{45}{c^{2}-1}$ or $a=\frac{15-15 c}{1-c}=\frac{50-5^{2}}{1-c^{2}}$ ..... 1 point
Writing the quadratic equation $c^{2}-5 c+4$ ..... 1 point
Solving the quadratic equation $c_{1}=4$ and $c_{2}=1$ ..... 1 point
Calculating the values $a=2, b=3$ ..... 1 point
Writing the function $f(x)=2+3 \cdot 4^{x}$ ..... 1 point