Show that the equation e^x = 1/2 + cos(2x) − 2 sin x has a UNIQUE solution in the interval 0, 􏰀π􏰁 /4 .

Question

Show that the equation e^x = 1/2 + cos(2x) − 2 sin x has a UNIQUE solution in the

interval 0, 􏰀π􏰁 /4 .

Answer 1

e^x = 1/2 + cos(2x) - 2sinx.

f(x) = e^x - 1/2 - cos(2x) + 2sinx

f(0) = -1/2 < 0

f(π/4) = e^π/4 - 1/2 + 2sinπ/4 = 1.72 > 0

By Intermediate value theorem f(x) has at least one zero on interval [0, π/4]

Now f'(x) = e^x + 2sin(2x) - 2cosx = e^x + 2sinx(2cosx - 1) > 0 on [0, π/4] so f(x) increasing on this interval and because zero is unique