max/min (1 Viewer)

[word.]

when proving a point is a maximum is the table enough?
sometimes you just know that it's going to be a max
so if you found a max at x = 3,
will just writing f'(3 - a) > 0, f'(3) = 0, f'(3 + a) < 0 get you full marks for the proving?

because sometimes there's 50 other variables and difficult to determine that it's a max

for example in my trial, which was ripped off cssa and other papers i think:

Q9a

A light is to be placed over the centre of a circle. The intensity I of the light varies as the sine of the angle (@) at which the rays strike the illuminated surface, divided by the square of the distance (d) from the light i.e. I = (ksin@)/d² where k is a constant.

i) show that I = ky/(y² + a²)^3/2)
don't worry about that one just, just for reference

ii) Find the best height for a light to be placed over the centre of a circle in order to provide maximum illumination to the circumference.

differentiating I you get a stat. point at y = a/Sqrt(2)
now would writing f'(a/Sqrt(2) - 1) > 0, f'(a/Sqrt(2)) = 0, f'(a/Sqrt(2) + 1) < 0 got me the mark for proving it was a max? because it sounds pretty stupid

 

[100percent]

sometimes it is easier to find f"(x) and sub the point in,
eg, if f"(x)>0 then it's min.
if f"(x)<0 then it's max

 

[word.]

yeah but not in this case however
2nd derivative will take very long and is very prone to error
but then again i'm pretty crap at differentiation...

 

[icycloud]

Is that the full question??

 

[rama_v]

I would always sub in a small number and write down the answer when u compute f'(x) so u can show the examiners that u just didn't magically assume that its negative at f'(3+a) and positive for f'(3-a).

But you can make it easier for yourself by usually disregarding teh denominator, for example question 10 (b) in 2002 HSC and also the last question in 2003 HSC, the denominator is always >0 so just check the numerator, sub in some values and say whether they are positive or negative. I hope you get what I mean

 

[Srixon]

just use second derivative it doesnt take no longer it is shorter if any thing

 

[Trippendicular]

Hey, does anyone happen to have copy of maximum and minimum problems like the onces found in Questions 9 and 10 from the HSC but not actually HSC questions cause I've done all the papers from the last 20 years. So just any from trials or whatever and preferably with worked solutions, thanks

 

[acmilan]

Sometimes second derivative is impractical, especially when the first derivative is so massive that the 2nd one will be even more massive, and thus making it difficult to prove if it will be positive or negative if all you have to go by is a letter.

I suggest if you do test using first derivative, dont go too far from the stationary point, sometimes going one unit to the left/right of the curve may result in a wrong answer depending on the shape of the curve. If the answer is completely numeric, + and - 0.1 or something around there would usually be pretty safe and always get you the marks.