See the attached graph plotted in Matlab.
a) approximate exterma
There is a maximum close to 2.
b)
>> f = (4/3)*x*sqrt(3-x)f =(4*x*(3 - x)^(1/2))/3
1. Asymptotes -
>> limit (f, inf)ans =Inf*1i>> limit (f, -inf)ans =-Inf
This means there is no horizontal asymptote.
Vertical asymptote is at x=3
2. >> f1 = diff(f)f1 =(4*(3 - x)^(1/2))/3 - (2*x)/(3*(3 - x)^(1/2))>> pretty(f1)sqrt(3 - x) 4 2 x------------- - ------------- 3 3 sqrt(3 - x)
We can find the ciritical points by finding the roots of the derivative of f.
>> crit_pts = solve(f1)crit_pts =2
There is only one maximum [-inf, inf] and it is at x = 2, and y = 2.667.
The results in b matches with results in a.
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