Let {{MATH_142}} be a continuous function on the real numbers and consider {{MATH_143}} From our previous work we know that {{MATH_144}} is increasing when {{MATH_145}} is positive and {{MATH_146}} is decreasing when {{MATH_147}} is negative. Moreover, with careful observation, we can even see that {{MATH_148}} is concave up when {{MATH_149}} is positive and that {{MATH_150}} is concave down when {{MATH_151}} is negative. Thinking about what we have learned about the relationship of a function to its first and second derivatives, it is not too hard to guess that there must be a connection between {{MATH_152}} and the function {{MATH_153}}. This is a good guess, check out our next theorem: