A student writes the following claim: "The function is continuous on the interval
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A student writes the following claim:
"The function h(x)=x1 is continuous on the interval [1,∞), which is closed at x=1 and extends without bound to the right. Therefore, by the Extreme Value Theorem, h must attain both a maximum and a minimum value on this interval."
(a) Identify the specific hypothesis of the Extreme Value Theorem that is violated in the student’s argument.
(b) Provide a corrected conclusion about the existence (or non-existence) of absolute extrema for h on [1,∞). Justify your response.
(c) Suppose the interval is changed to [1,4]. Determine whether h attains its absolute maximum and minimum on this new interval, state those values, and justify your conclusions.