A student writes the following claim.
"Let f be a function defined on the closed interval [a,b] that is differentiable on (a,b). If c is a point in (a,b) satisfying f′(c)=0 and f′′(c)>0, then f attains its absolute minimum at c on the interval [a,b]."
Answer the prompts that follow to evaluate the validity of the student’s claim.
(a) Identify and describe
i. a hypothesis that is stated in the claim but is not sufficient by itself to guarantee the conclusion, and
ii. a hypothesis that is missing from the claim yet is essential for the conclusion to hold.
(b) Construct a specific function g and a closed interval [p,q] for which g′(d)=0 and g′′(d)>0 at a point d∈(p,q) but at which g(d) is
i. not the smallest value that g attains on [p,q], or
ii. not attained on the interval at all.
Clearly state which omitted hypothesis your example violates and explain why your example contradicts the student’s conclusion.
(a) Identify and describe
i. a hypothesis that is stated in the claim but is not sufficient by itself to guarantee the conclusion, and
ii. a hypothesis that is missing from the claim yet is essential for the conclusion to hold.
(b) Construct a specific function g and a closed interval [p,q] for which g′(d)=0 and g′′(d)>0 at a point d∈(p,q) but at which g(d) is
i. not the smallest value that g attains on [p,q], or
ii. not attained on the interval at all.
Clearly state which omitted hypothesis your example violates and explain why your example contradicts the student’s conclusion.
(c) For a function that satisfies all necessary hypotheses you listed in part (a), explain why the student’s conclusion is in fact correct. Your explanation should include logical reasoning that connects the hypotheses to the desired conclusion without appealing to any results beyond first–year calculus.