Question

Calculus, multiple choice:
Find the absolute extrema of f(x)= (x^3 -4x^2 +7x)/x on the interval (0,3]
Help is very much appreciated!

Answer 1

maximum: none, minimum: 4

maximum: 7, minimum: 4

maximum: 7, minimum: 3

maximum: none, minimum: 3

none of these

Answer 2

these are my choices. I think the fact that this is not a completely closed interval is getting to me. how should I start?

Answer 3

i'll just use the quotient rule to find derivative and then see where that goes... give me a sec.

Answer 4

derivative simplifies to f'= 2x-4 ?

Answer 5

yes. It would have been ok to simplify the original equation to
x^2 -4x + 7

and f' = 2x -4 = 0

Answer 6

ok I went around my elbow. :3
so f'=0 when x=2...
ah, what do I do next?

Answer 7

plug it into f ? f(2)= 3

Answer 8

one way to go is evaluate the original function at x=2
compare to the value of the function at x=3
(and at x=0, though 0 is not in the domain... it gives us an idea of the shape of the curve)

Answer 9

concave upward parabola in upper two quadrants is what I am getting

Answer 10

so there would be no maximum, only a minimum

Answer 11

sounds correct. the value at x=3 is 4
the value at x=0 is 7 (but we never reach 7 because 0 is excluded from the domain.)
so only a min exists in the interval (0,3]

Answer 12

so, with these answer choices would it be A? because the minimum is at y=4?

Answer 13

we can cross out B and C easily, and E too, I believe...
A vs. D is tricky and I'm not entirely sure which one it is asking for

Answer 14

what happened to
plug it into f ? f(2)= 3

Answer 15

what? oh, sorry I was confused by your mention of (3,4) being a point.
so (2,3) is a point, too. so the minimum is y= 3 and the answer is D

Answer 16

yes, no max, min of 3 (at x=2)

Answer 17

ok thank you very much! I had just been worried that the interval was a 'hybrid' of sorts, but now I know that you can solve the same way! :)
you get a medal, and i'll close the question.
again, thanks for your time!