It is widely accepted that the classroom teaching practices of in-service secondary mathematics teachers can benefit from the advanced mathematics they learned during their university education. However, pinpointing explicit instances of how this advanced knowledge informs teaching practices has proven challenging for both teachers and mathematicians alike. This paper aims to address this challenge by examining three statements related to Calculus and demonstrating how advanced mathematical knowledge can be utilized to generate counterexamples. By using results encountered in university-level courses, we construct functions that counter these statements. In conclusion, we identify a thought process that can be generally followed in assessing whether a given statement is true, highlighting the explicit connections of advanced mathematics in generating counterexamples via these statements. Similar endeavors of carefully generating counterexamples using advanced mathematics in other areas of mathematics can contribute to the designing of courses for teacher development that aim at explicitly highlighting connections between university mathematics and school mathematics.