Discuss the curve f(x)= 1-3x+5x^2-x^3 with respect to concavity, points of inflection, and local maxima and minima. Use this information to sketch the curve.
2006-11-08 15:19:35 · 4 answers · asked by Lauren K 1 in Science & Mathematics ➔ Mathematics
f'(x)=-3+10x-3x^2
f''(x)=10-6x
max and min are where f'(x)=0
3x^2-10x+3=0
(x-3)(3x-1)=0
x=3,1/3
to see if they are max or min use f''(x)
f''(3)=-8 means concave down= max at x=3
f''(1/3)=8 means concave up= min at x=1/3
inflection points are where f''(x)=0
10-6x=0
x=5/3
f''(0)=10 means concave up on the interval (-inf,5/3)
f''(5)=-20 means concave down on the interval (5/3,inf)
2006-11-08 15:31:54 · answer #1 · answered by Greg G 5 · 0⤊ 0⤋
ok be attentive to the inflection factor is the factor the place the function shift from the +ive to -ive or the alternative . so iy's like the max/mini factors . and on the topic of the 2d answer you're precise , u discover the mini and max from the 1st derivative and u equivalent it to 0 and after that u take any factor on the ultimate suited and the strengthen if it replaced into increasing and reducing then it is max and the alternative it is mini answer f '(x)= 3x^2 + x -2 =0 ==> (3x-2)(x+a million)=0 ==> x=2/3 and x=-a million so we now take any quantity till now 2/3 and after 2/3 f '(0)= 3(0)^2 + 0 -2 = -2 so the -ive is recommend that the unique function is reducing f '(a million)= 3(a million)^2 + a million -2 = 2 so the +ive is recommend that the unique function is increasing so there is at x=2/3 interior reach mini fee for x=-a million f ' (-2) = 3(-2)^2 -2 -2 = 8 so the +ive is recommend that the unique function is increasing f '(0)= 3(0)^2 + 0 -2 = -2 so the -ive is recommend that the unique function is reducing so there is at x=-a million interior reach max fee
2016-12-14 04:05:55 · answer #2 · answered by binford 4 · 0⤊ 0⤋
im just going to give you the answers...if you want an explanation email me jorgealbertor@hot...
f'(x) = -3 + 10x - 3x^2
x = 1/3 , 3
rel. min.: ( 3 , 10 )
rel. max.: ( 1/3 , 14/27 )
f''(x) = 10 - 6x
x = 5/3
concave up: ( - infinity , 5/3 )
concave down: ( 5/3 , + infinity )
inflection points: ( 5/3 , 142/27 )
2006-11-08 15:40:03 · answer #3 · answered by Jorge. 2 · 0⤊ 0⤋
take derivative of f(x)
f'(x) = -3x^2 + 10x -3
0 = f'(x) solve for x
check value left and right of your solution you would know it it inflection max or min
2006-11-08 15:29:09 · answer #4 · answered by The Clueless Philospher 2 · 0⤊ 0⤋