Number of solutions

Shrey

The number of solutions of the equation sin[{pi*x/2(root3)}]=x^3-2(root3)x+4 is?

Manasi

the LHS is a sine function, hence, the range of LHS = [-1,1]

for a solution, the graph of RHS should intersect the graph of sine function of the LHS,

which is possible only if

for some real 'x', -1=< RHS <= 1

now for x>0 , RHS > 0

also for x = -sqrt(3), LHS=0 and RHS = 1.9

x = - 3 sqrt(3) /2 , LHS=1 and RHS = -8.5 (approx)

i.e. the polynomial in the RHS changes its sign for some x in the range, ( - 3 sqrt(3) /2 , sqrt(3) ) ...

Thus there exist only one solution to the equation