6.27. Write the equation of the tangent line to the graph of the function $f(x)=\left(x^{5}-1\right) / 5$ at the point of its intersection with the x-axis.
Solution. Let's find the coordinates of the point of intersection of the curve with the x-axis: $f(x)=\left(x^{5}-1\right) / 5=0, x_{0}=1, y_{0}=0$. At this point, the derivative $f_{x}^{\prime}(x)=x^{4}$ is equal to $f_{x}^{\prime}(1)=1$. The equation of the tangent line is $y=x-1$.