How do you use the Intermediate Value Theorem to show that the polynomial function sin2x + cos2x - 2x = 0sin2x+cos2x2x=0 has a zero in the interval [0, pi/2]?

1 Answer

Refer to explanation

Explanation:

First set f(x)=sin2x+cos2x-2xf(x)=sin2x+cos2x2x then we have that

f(0)=sin0+cos0-2*0=cos0=1>0f(0)=sin0+cos020=cos0=1>0

and

f(pi/2)=sin(pi/2)+cos(pi/2)-2*(pi/2)=1+0-pi=1-pi<0f(π2)=sin(π2)+cos(π2)2(π2)=1+0π=1π<0

Because f(0)*f(pi/2)<0f(0)f(π2)<0 the IVT tells us that f(x)f(x)

has a zero at [0,pi/2][0,π2]