Step 1: Understand the situation.
A convex lens forms an image that is one fourth the size of the object. We must find the object distance in terms of the focal length $f$.
Step 2: Use the magnification rule.
Magnification is $m=\frac{v}{u}$. A convex lens making a small image gives a real, inverted image, so we take $m=-\frac{1}{4}$.
Step 3: Express $v$ in terms of $u$.
From $\frac{v}{u}=-\frac{1}{4}$ we get $v=-\frac{u}{4}$.
Step 4: Write the lens formula.
The lens equation is \[ \frac{1}{v}-\frac{1}{u}=\frac{1}{f}. \]
Step 5: Substitute $v=-\frac{u}{4}$.
\[ \frac{1}{-\frac{u}{4}}-\frac{1}{u}=\frac{1}{f}\Rightarrow -\frac{4}{u}-\frac{1}{u}=\frac{1}{f}. \]
Step 6: Combine and solve.
\[ -\frac{5}{u}=\frac{1}{f}\Rightarrow u=-5f. \]
Step 7: Read the size.
The minus sign just means the object sits in front of the lens. The distance is $5f$, which is option (2).
\[ \boxed{5f} \]