Answer to Question #160024 in Real Analysis for Luna

Question

For which function is given below, there are is a maximum and a minimum values on the given interval?

a) f(x)=x² on (0,1)

b) f(x)=1/x if x=0 on [0,1] , 0 if x=0

c) f(x)=2x+1 on R

d) f(x)=x²+1/x on [1,2]

e) None of the above.

Answer 1

a) No. $f(x)=x^2$ increases on open interval $(0, 1)$ and is undefined at $x=0$ and $x=1.$

b) No. $f(x)=1/x$ decreases on open interval $(0, 1)$ from $\infin$ to $1.$

$f(x)$ has minimum value on $[0, 1],$ but has no maximum value on $[0, 1].$

c) No. $f(x)=2x+1$ increases from $-\infin$ to $\infin$ on $\R.$

d) Yes. The function $f(x)=x^2+1/x$ is defined on $[1, 2].$ $f'(x)=2x-1/x^2$

$f'(x)>0$ on $[1, 2]$ => $f(x)$ increases on $(1, 2)$

$f(1)=2, f(2)=9/2$

The function $f(x)=x^2+1/x$ has the maximum with value of $9/2$ on $[1, 2]$ at $x=2.$

The function $f(x)=x^2+1/x$ has the minimum with value of $2$ on $[1, 2]$ at $x=1.$

d) $f(x)=x^2+1/x$ on $[1, 2].$