Question

f(x) = x4 - 50x2 + 4
find the intervals on which f is increasing and decreasing. local min and max values. inflection points. intervals on which f is concave up and concave down.

Answer 1

1st and 2nd derivatives give you all that information

Answer 2

y' = 4x^3 -100x

y'' = 12x^2

Answer 3

when y'=0 we get some critical points:

y' = 4x^3 - 100x

x(4x^2 - 100) = 0

x(2x-10)(2x+10) = 0

x = 0, x=5, x=-5 are all critical points to check

<.......(-5)......(0).......(5).......>

Answer 4

y'' = 12x^2

12(0)^2=0 ; good candidate for inflection

12(-5)^2 = 12(25) which is (+), this is a MIN
12(5)^2 = 12(25) which is (+), this is ALSO a MIN

<.......(-5).......(0)........(5).......>
------0++++(0)++++0----->

This thing is a big "W" looking graph with -5, 0, 5 being the low, high, low respectively

Answer 5

(-inf,0) concave up
(0,inf) concave up
(-5,5) concave down :) depends on where you lookat it from I suppose

Answer 6

dont try to make any sense outta me line graph; it messed up anywhoos :)

Answer 7

decreasing (-inf,-5)
increaseing (-5,0)
decreasing (0,5)
increasing (5,inf)

Answer 8

any of this making sense?

Answer 9

f(-5) = (-5)^4 - 50(-5)^2 + 4 = (-5,-621) MIN
f(0) = 0 - 0 + 4 = (0,4) MAX
f(5) = (5)^4 - 50(5)^2 + 4 = (5,-621) MIN

Answer 10

that should be all the answers :)