I am studying for a test and really need to know some examples of function with upper bound and lower bounds. I hope someone can be kind enough to help.Thank you
Please give examples of function defined at all points on the interval
a. $[3,5]$ have upper bound and lower bound but no minimum value
b. $[3,∞]$ continuous with upper and lower bound but no minimum value
c. $(3,5]$ continuous with upper bound and lower bounds but no min value
Best Answer
For part a).take: $$f(x) = \begin{cases} -x+6 \text{ if}\text{ } 3 \leq x < 4\\ 4 \text{ if} \text{ }x = 4\\ x-2 \text{ if} \text{ }4 < x \leq 5\end{cases}$$.
For part b), take: $$f(x) = \begin{cases} -x+6 \text{ if}\text{ } 3 \leq x < 4\\ 4 \text{ if} \text{ }x = 4\\ x-2 \text{ if} \text{ } 4 < x \leq 5\\ 7 \text{ if} \text{ } x > 5\end{cases}$$.
For part c), take: $f(x) = x$.