Original
Let {{MATH_236}}, for {{MATH_237}}. Locate the global extremums
of {{MATH_238}} on the closed interval {{MATH_239}}. Does the function {{MATH_240}} satisfy
the conditions of the Extreme Value Theorem on its domain?
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{{CMD_242}}Yes, because {{MATH_243}} is continuous on {{MATH_244}}.{{CMD_245}}
{{CMD_246}}Yes, because {{MATH_247}} is continuous on {{MATH_248}}.{{CMD_249}}
{{CMD_250}}No, because {{MATH_251}} is not continuous on {{MATH_252}}.{{CMD_253}}
{{CMD_254}}No, because {{MATH_255}} is not differentiable on {{MATH_256}}.{{CMD_257}}
{{CMD_258}}
Therefore, the Extreme Value Theorem guarantees that the function {{MATH_259}}
attains both global extremums on its domain. The global extremums
occur at the end points or at critical points.
Find the critical points of {{MATH_260}}. First, compute the derivative of {{MATH_261}}.
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{{MATH_263}}
{{CMD_264}}
In order to find the critical points of {{MATH_265}}, we have to solve the
equation
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{{MATH_267}}
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It follows that the function {{MATH_269}} has only one critical point
{{MATH_270}}. Find {{MATH_271}}.
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{{MATH_273}}
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In order to locate the global extremums of {{MATH_275}}, we have to evaluate
{{MATH_276}} at the end points and at the critical point.
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{{MATH_278}}
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{{MATH_281}}
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{{MATH_284}}
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Order the three values, {{MATH_286}}, {{MATH_287}}, and {{MATH_288}}, from smallest to largest. You should replace {{MATH_289}} with its value, when you write {{MATH_290}} in your answer below.
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{{MATH_292}}
{{CMD_293}}
Based on this comparison, find the location of the global minimum and global maximum of {{MATH_294}}. Circle the correct answer.
{{CMD_295}}
{{CMD_296}}{{MATH_297}} has a global maximum at {{MATH_298}} and global minimum at {{MATH_299}}.{{CMD_300}}
{{CMD_301}}{{MATH_302}} has a global maximum at {{MATH_303}} and global minimum at {{MATH_304}}.{{CMD_305}}
{{CMD_306}} {{MATH_307}} has a global maximum at {{MATH_308}} and global minimum at {{MATH_309}}.{{CMD_310}}
{{CMD_311}}{{MATH_312}} has a global maximum at {{MATH_313}} and global minimum at {{MATH_314}}.{{CMD_315}}
{{CMD_316}}{{MATH_317}} has a global maximum at {{MATH_318}} and global minimum at {{MATH_319}}.{{CMD_320}}
{{CMD_321}}{{MATH_322}} has a global maximum at {{MATH_323}} and global minimum at {{MATH_324}}.{{CMD_325}}
{{CMD_326}}
{{CMD_327}}