Find the minimum value of #f(x)=(x^2+1/x)/(x^2-(1-1/x^2)/(1/x+1/x^2)# over the interval #1 le x le 2#. Write answer as *exact* decimal?

3 Answers
Jan 6, 2018

Minimum value #f(2) = 1.5#

Explanation:

Given:

#f(x) = (x^2+1/x)/(x^2-(1-1/x^2)/(1/x+1/x^2)#

#color(white)(f(x)) = (x^2+1/x)/(x^2-(x^2-1)/(x+1)#

#color(white)(f(x)) = (x^2+1/x)/(x^2-(x-1))#

#color(white)(f(x)) = (x^3+1)/(x(x^2-x+1))#

#color(white)(f(x)) = ((x+1)(x^2-x+1))/(x(x^2-x+1))#

#color(white)(f(x)) = (x+1)/x#

#color(white)(f(x)) = 1+1/x#

with excluded value #x != -1#

Note that #1/x# is monotonically decreasing in the interval #[1, 2]#.

So the minimum value is attained when #x=2#:

#f(2) = 1 + 1/2 = 1.5#

Jan 6, 2018

#3/2#

Explanation:

#"Multiply numerator and denominator by (1/x + 1/x²)."#
#"Then we obtain f(x) = : "#

#((x^2 + 1/x)(1/x + 1/x^2)) / ((x^2(1/x + 1/x^2) + 1/x^2 - 1)#
#= (x + 1 + 1/x^2 + 1/x^3)/(x + 1 + 1/x^2 - 1)#

#"Now multiply numerator and denominator by x³ : "#

#= (x^4 + x^3 + x + 1)/(x^4 + x)#

#"Now derive using the quotient rule."#

#f'(x) = ((4x^3+3x^2+1)(x^4+x)-(x^4+x^3+x+1)(4x^3+1))/(x^4+x)^2#

#"The derivative is zero when the numerator is zero :"#

#- x^6 - 2 x^3 - 1 = 0#
#=> (x^3 + 1)^2 = 0#
#=> x^3 + 1 = 0#
#=> x = -1#

#"There is a problem though as the denominator is also zero"#
#"for this value of x, so we have the case 0/0, so we must"#
#"divide away the common factor (x+1)^2 :"#

#f'(x) = -(x^3+1)^2 / (x^2(x^3+1)^2)#
#= -1/x^2#

#"So the derivative is always negative. This means that the"#
#"function is ever decreasing, so the minimum over the interval"#
#"[1, 2] is reached for x=2 :"#

#f(2) = (2^4 + 2^3 + 2 + 1)/(2^4 + 2)#
# = 27/18#
# = 3/2#

Jan 6, 2018

Minimum value is #1.5#

Explanation:

#f(x)=(x^2+1/x)/(x^2-(1-1/x^2)/(1/x+1/x^2)#

= #((x^3+1)/x)/(x^2-(x^2(1-1/x^2))/(x^2(1/x+1/x^2))#

= #((x^3+1)/x)/(x^2-(x^2-1)/(x+1)#

= #((x^3+1)/x)/(x^2-(x-1)#

= #((x+1)(x^2-x+1))/x xx1/(x^2-x+1)#

= #(x+1)/x=1+1/x#

Observe that at #x=1# we have #f(x)=2# and as #x# increases to #2#, the value of #1/x# comes down

and it is minimum wheen #x=2# and it is #1+1/2=1.5#

graph{(x^2+1/x)/(x^2-(1-1/x^2)/(1/x+1/x^2) [-0.983, 4.017, 0.23, 2.73]}