Question #160024

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.


1
Expert's answer
2021-02-04T08:02:45-0500

a) No. f(x)=x2f(x)=x^2 increases on open interval (0,1)(0, 1) and is undefined at x=0x=0 and x=1.x=1.


b) No. f(x)=1/xf(x)=1/x decreases on open interval (0,1)(0, 1) from \infin to 1.1.

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


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


d) Yes. The function f(x)=x2+1/xf(x)=x^2+1/x is defined on [1,2].[1, 2]. f(x)=2x1/x2f'(x)=2x-1/x^2

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

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

The function f(x)=x2+1/xf(x)=x^2+1/x has the maximum with value of 9/29/2 on [1,2][1, 2] at x=2.x=2.

The function f(x)=x2+1/xf(x)=x^2+1/x has the minimum with value of 22 on [1,2][1, 2] at x=1.x=1.


d) f(x)=x2+1/xf(x)=x^2+1/x on [1,2].[1, 2].




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