Show that an equation has a solution

Question

show that (x^3)/2=sin [ πx/4 + π/2) has a solution over the interval [0,2] 

Answer 1

 

Let f(x) = sin [ pi*x/4 + pi/2] - x^3/2

 

f(0) = sin(pi/2)  = 1

 

f(1) = (sqrt(2)-1)/2 >0

 

f(2) = -4

 

Since it is a continuous function and the sign changes from positive to

negative, there must be a root inside that interval.

 

Per the bisection method it is in fact approximately:

1.0933113104292560*