anonymous
  • anonymous
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.
Mathematics
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SOLVED
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katieb
  • katieb
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amistre64
  • amistre64
1st and 2nd derivatives give you all that information
amistre64
  • amistre64
y' = 4x^3 -100x y'' = 12x^2
amistre64
  • amistre64
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).......>

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amistre64
  • amistre64
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
amistre64
  • amistre64
(-inf,0) concave up (0,inf) concave up (-5,5) concave down :) depends on where you lookat it from I suppose
amistre64
  • amistre64
dont try to make any sense outta me line graph; it messed up anywhoos :)
amistre64
  • amistre64
decreasing (-inf,-5) increaseing (-5,0) decreasing (0,5) increasing (5,inf)
amistre64
  • amistre64
any of this making sense?
amistre64
  • amistre64
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
amistre64
  • amistre64
that should be all the answers :)

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