Find the absolute minimum value of
f(x) =(3/2)x + 2/(9x^2)
on (0 ,∞).

Question

Find the absolute minimum value of
f(x) =(3/2)x + 2/(9x^2)
on (0 ,∞).

hero · Answer

I suppose graphing it is out of the question

hero · Answer

I found the minimum value. Are you there?

anonymous · Answer

i try taking the derivative then i set that equal to zero but then i can never get past that

hero · Answer

f(x) = (3/2)x + 2/(9x^2)
f'(x) = 3/2 - 4/(9x^3)
0 = 3/2 - 4/(9x^3)
4/(9x^3) = 3/2

Cross multiply:
2(4) = 3(9x^3)
8 = 27x^3
8/27 = x^3
cube root(8/27) = x
2/3 = x

Plug x = 2/3 back into f(x):
f(2/3) = (3/2)(2/3) + 2/(9(2/3)^2)
f(2/3) = 1 + 2/(9(4/9))
f(2/3) = 1 + 2/4
f(2/3) = 1 + 1/2
f(2/3) = 2/2 + 1/2
f(2/3) = 3/2

Thus the minimum value of f(x) is 3/2

hero · Answer

If you have any questions about any of that, let me know

anonymous · Answer

OHHHHH, ok i had gotten to cube root(8/27) then thats where i got stuck! That makes sense, Thankyou so much! :]