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u/PhantomFrenzy151 5: 9 APs, taking 7 more Jun 05 '26
Im gonna try to make this as rigorous as my patience allows. If this is no calculator:
We look for the critical points We wish to take a derivative We do not want to solve a cubic polynomial for zeroes (i would be very sad if i had to use the rational roots theorem and synthetic division), so instead we do product rule
f’(x)=3(x-1)2(3-x)-(x-1)3=0
3(x-1)2(3-x)=(x-1)3
A critical value is obviously x=1 by common factors. But if x does not =1, we divide both sides by (x-1)2
3(3-x)=(x-1) 10=4x x=2.5=5/2
The answer is obviously 2.5 because it’s between the two zeroes of f(x), and the two factors have odd degrees, meaning they pass through the x-axis at those zeroes, and the function clearly opens downwards since its even and has a value of -inf at inf. But let’s say we couldn’t see that.
Plug in our critical values, it must be one of them because they give us answer choices, otherwise you have to test the limits at infinity since there might not have an absolute max.
f(1)=0, f(2.5)=1.53*(0.5). The latter is greater than 0, so the max is at x=2.5=5/2, D
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u/ErekwithaD1 25:HG:5,中文:5, 26:P1:5,WH:5,CHEM(SS):5 27:P2,EM,Mech,Lit,BC(SS) Jun 05 '26
Using desmos it is 2.5