i have already completed the question im just using this as an example
For question c i found f'(x) then i found the value of x via simple algebra manipulation
but what if the question asks for the maxmimum value of P??
If you have the value of 'x' that minimizes P...... Sub that value in to the equation for P to find the minumum value of P.
P = 100x^(-1) + x ( pi + 8 - 2sqrt(3)/ 4 take the derivative, set to 0
P' = -100x^(-2) + ( pi + 8 - 2sqrt (3) ) / 4 = 0 multiply through by x^2
[ ( pi + 8 - 2sqrt (3) / 4 ] x^2 - 100 = 0
[ pi + 8 - 2sqrt (3) ] x^2 = 400
x^2 = 400 / [ pi + 8 - 2sqrt (3) ] take the positive root
x = 7.218
Sub this back into P to find the minimum P ≈ 27. 7 m
We can verify that this is the minimum ........
P " = 200/ x^3
Subbing in x = 7.218 will produce a positive....thus...x = 7.218 produces a minimum value