Question

Let f be a function with second derivative given by f''(x)=sin(2x)-cos(4x). How many points of inflection does the graph of f have on the interval [0,10]?

(A)Six
(B)Seven
(C)Eight
(D)Ten
(E)Thirteen

Answer 1

for f"=0 you need
sin2x - cos4x = 0
sin2x - (1 - 2sin^2(2x)) = 0
2sin^2(2x) - sin(2x) + 1 = 9
(2sin2x+1)(sin2x-1) = 0
sin2x = -1/2
sin2x = 1

Count the solutions in the interval. Looks like ten to me.

Answer 2

You can find POI when 2nd derivative = 0. Calculator should show the graph with the number of times when y=0. The tricky part is to trace the graph to make sure y=0. (it may look like that on the calculator, but it's not y=0)