Let f be the function with the derivative given by f'(x)=sin(x^2+1). How many relative extrema does f have on the interval 2<X<4?

Question

The answer can only range from 1-5

Answer 1

Sin(T) = 0 when T=0 or T=pi, or integer multiples thereof...

x^2 + 1 = 0 ---> x = +or- i

x^2 + 1 = pi --> x = +or- sqrt( pi - 1)

unfortunately, sqrt(pi-1) is NOT in the domain 2 < x < 4

the next place it happens is when T=2*pi,

which forces x^2 + 1 = 2*pi

x = +or- sqrt( 2*pi-1) which is approximately 2.29

the next place it happens is when T = 3 * pi

which forces x^2 +1 = 3*pi

x = +or- sqrt( 3*pi - 1) which is approximately 2.9

the next place it happens is when T= 4*pi

which forces x^2 + 1= 4 * pi

x = +or- sqrt( 4 * pi - 1) which is approximately 3.4

the next place it happens is when T= 5*pi

which forces x^2 +1 = 5 *pi

x = +or- sqrt(5*pi-1) which is approximately 3,83

the next place it happens is when T=6*pi

which forces x^2 +1 6 * pi

x = +or- sqrt(6*pi - 1) which is approximately 4.22

So there are 4 flex points:

x = sqrt(N*pi-1) where N=2,3,4,5