Question

The number of real solutions to the system of equations y=2sin(x^2) and y=x^3-x is

a. 0
b. 1
c. 2
d. 3

Please show your work! Thank you!

Answer 1

Let
f1(x)=2sin(x^2)
and
f2(x)=x^3-x
We look for the number of solutions for which
f1(x)=f2(x).

Since the values of f1(x) lies between ±2, we only have to limit our search for values of x where f2(x)≤2, or approximately [-1.5,1.5].

By inspection, we see that x=0 is a solution.

f1'(x)=4xcos(x^2), and
f1'(0)=0
f2'(x)=3x^2-1
f2'(0)=-1

Since f1'(0)>f2'(0), and we know that
Lim f2(x) = -∞
x->-∞
and
Lim f2(x) = ∞
x->∞
There must be one more solution for x<0 and for x>0.

There answer is therefore 3 solutions.