# Question Which of the following statements are true and which are false? I. An absolute maximum of a function $$f(x)$$ is the largest value of all the $$y$$-values of $$f(x)$$ (on the entire domain of $$f(x)$$ ). II. A function can have more than one absolute maximum. III. A local maximum $$f(c)$$ of a function $$f(x)$$ is the largest of all $$y Which of the following statements are true and which are false? I. An absolute maximum of a function \( f(x)$$ is the largest value of all the $$y$$-values of $$f(x)$$ (on the entire domain of $$f(x)$$ ). II. A function can have more than one absolute maximum. III. A local maximum $$f(c)$$ of a function $$f(x)$$ is the largest of all $$y$$-values of $$f(x)$$ when $$x$$ is near $$c$$ on both sides (specifically, on some open interval containing $$c$$ ). IV. A function can have more than one local maximum. V. A function can not have both a local and an absolute maximum at the same point $$c$$. $$\mathrm{VI}$$. It is possible for a function to have $$\mathrm{NO}$$ absolute maximum and $$\mathrm{NO}$$ absolute minimum. VII. If a function is CONTINUOUS on a CLOSED BOUNDED interval, then the function is guaranteed to have an absolute maximum and an absolute minimum. VIII. The Extreme Value Theorem (Theorem 4.1) says that only continuous functions have an absolute maximum and an absolute minimum on every closed bounded interval.

Transcribed Image Text: Which of the following statements are true and which are false? I. An absolute maximum of a function $$f(x)$$ is the largest value of all the $$y$$-values of $$f(x)$$ (on the entire domain of $$f(x)$$ ). II. A function can have more than one absolute maximum. III. A local maximum $$f(c)$$ of a function $$f(x)$$ is the largest of all $$y$$-values of $$f(x)$$ when $$x$$ is near $$c$$ on both sides (specifically, on some open interval containing $$c$$ ). IV. A function can have more than one local maximum. V. A function can not have both a local and an absolute maximum at the same point $$c$$. $$\mathrm{VI}$$. It is possible for a function to have $$\mathrm{NO}$$ absolute maximum and $$\mathrm{NO}$$ absolute minimum. VII. If a function is CONTINUOUS on a CLOSED BOUNDED interval, then the function is guaranteed to have an absolute maximum and an absolute minimum. VIII. The Extreme Value Theorem (Theorem 4.1) says that only continuous functions have an absolute maximum and an absolute minimum on every closed bounded interval.
Transcribed Image Text: Which of the following statements are true and which are false? I. An absolute maximum of a function $$f(x)$$ is the largest value of all the $$y$$-values of $$f(x)$$ (on the entire domain of $$f(x)$$ ). II. A function can have more than one absolute maximum. III. A local maximum $$f(c)$$ of a function $$f(x)$$ is the largest of all $$y$$-values of $$f(x)$$ when $$x$$ is near $$c$$ on both sides (specifically, on some open interval containing $$c$$ ). IV. A function can have more than one local maximum. V. A function can not have both a local and an absolute maximum at the same point $$c$$. $$\mathrm{VI}$$. It is possible for a function to have $$\mathrm{NO}$$ absolute maximum and $$\mathrm{NO}$$ absolute minimum. VII. If a function is CONTINUOUS on a CLOSED BOUNDED interval, then the function is guaranteed to have an absolute maximum and an absolute minimum. VIII. The Extreme Value Theorem (Theorem 4.1) says that only continuous functions have an absolute maximum and an absolute minimum on every closed bounded interval.
&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/8We need to check True and False .I. True, An absolute maximum of a function f(x) is the largest value of all the y-values of f on the domain of f .Explanation:Please refer to solution in this step.Step2/8II. False, A function has one absolute maximum if exist , since we have absolute maximum of f is \begin{align*} \mathrm{{M}} &\ge \mathrm{{y},\forall{x}\Rightarrow{y}={f{{\left({x}\right)}}}.} &\text{(1)} \end{align*}So Absolute maximum is unique if it is exist.Explanation:Please refer to solution in this step.Step3/8III. True, A local maximum $$\mathrm{{f{{\left({c}\right)}}}}$$of function $$\mathrm{{f{{\left({x}\right)}}}}$$ is the values as $$\mathrm{{f{{\left({c}\right)}}}\ge{y},\forall{x}}$$ such that $$\mathrm{{x}\in{\left({c}-\delta,{c}+\delta\right)}}$$ for small number \( ... See the full answer