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I am looking more for a better method of solving this exam question as opposed to a solution (already solved but in one area I believe, badly.)

f(x) =12lnx - x^(3/2)

the graph cuts the x-axis at A reaches a maximum point at C then cuts again at B.

A) show by calculation that the x coordinate at A lies between 1.1 and 1.2

b) b lies in the interval (n,n+1) where n is an integer. Determine B

c) find the value of x for which dy/dx = 0 and hence find the maximum value of C

d) find the range of values of x for which f(x) is increasing

c) and d) are pretty straight forward...

the only way I could calculate a) was to start by saying...lnx>1 (as with a lower value... x^(3/2) would be less than 1..lnx for x<1 yields a negative result) using a value of 1 for x I found that lnx = 1^(3/2)/12 and this gives a value of 1.087...x must now be greater than this number, repeating this method I found x around 1.101068...

question b) is my problem... how can the x position at B be calculated? the above method fails. (I only solved it by throwing a few numbers at it until I hit the correct range...I know from question c that I'm looking for a value greater than 4)

my knowledge so far is limited to differentiating and integrating polynomials and lnx, finding areas and stationery values etc...I haven't covered much more so far and have certainly not done any work on soving these types of problems...there must surely be a solution however.

f(x) =12lnx - x^(3/2)

the graph cuts the x-axis at A reaches a maximum point at C then cuts again at B.

A) show by calculation that the x coordinate at A lies between 1.1 and 1.2

b) b lies in the interval (n,n+1) where n is an integer. Determine B

c) find the value of x for which dy/dx = 0 and hence find the maximum value of C

d) find the range of values of x for which f(x) is increasing

c) and d) are pretty straight forward...

the only way I could calculate a) was to start by saying...lnx>1 (as with a lower value... x^(3/2) would be less than 1..lnx for x<1 yields a negative result) using a value of 1 for x I found that lnx = 1^(3/2)/12 and this gives a value of 1.087...x must now be greater than this number, repeating this method I found x around 1.101068...

question b) is my problem... how can the x position at B be calculated? the above method fails. (I only solved it by throwing a few numbers at it until I hit the correct range...I know from question c that I'm looking for a value greater than 4)

my knowledge so far is limited to differentiating and integrating polynomials and lnx, finding areas and stationery values etc...I haven't covered much more so far and have certainly not done any work on soving these types of problems...there must surely be a solution however.

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