Would this theorem hold if we were working on a closed interval {{MATH_183}}, but a function is not continuous on {{MATH_184}}? {{CMD_185}} {{CMD_186}} {{CMD_187}}yes{{CMD_188}} {{CMD_189}}no{{CMD_190}} {{CMD_191}} {{CMD_192}} Consider a function {{MATH_193}} on a closed interval {{MATH_194}}, defined by {{MATH_195}} for {{MATH_196}} and {{MATH_197}} . Does this function achieve its maximum or minimum? {{CMD_198}} {{CMD_199}} {{CMD_200}} Assume that a function {{MATH_201}} is continuous on a closed interval {{MATH_202}}. By the Extreme Value Theorem, {{MATH_203}} attains both global extremums on the interval {{MATH_204}}. How can we locate these global extrema? We have seen that they can occur at the end points or in the open interval {{MATH_205}}. If a global extremum occurs at a point {{MATH_206}} in the open interval {{MATH_207}}, then {{MATH_208}} has a local extremum at {{MATH_209}}. That means that {{MATH_210}} has a critical point at {{MATH_211}}. So, the global extrema of a function {{MATH_212}} occur either at the end points, {{MATH_213}} or {{MATH_214}}, or at critical points. If we want to locate the global extrema, we have to evaluate the function at the end points and at critical points, and compare the values.