Assume that a function {{MATH_630}} has a critical point at {{MATH_631}} and that {{MATH_632}}. If {{MATH_633}} happens to be decreasing on some interval containing {{MATH_634}}, then it changes from positive to negative at {{MATH_635}}. Therefore, if {{MATH_636}} is negative on some interval that contains {{MATH_637}}, then {{MATH_638}} is definitely decreasing, so there is a local maximum at the point in question. On the other hand, if {{MATH_639}} is increasing, then it changes from negative to positive at {{MATH_640}}. Therefore, if {{MATH_641}} is positive on an interval that contains {{MATH_642}}, then {{MATH_643}} is definitely increasing, so there is a local minimum at the point in question. We summarize this as the {{CMD_644}}second derivative test{{CMD_645}}.