- Can we use the IVT to conclude that
*f*(*x*) = *x*^{3} + 2*x* + 1 passes through *y* = 0 on the interval (-2,2)?

Answer

Yes. For starters, *f* is continuous on the closed interval [-2,2]. We have *a* = -2 and *b* = 2, therefore

Since *f*(*a*) = -11 < 0 < 13 = *f*(*b*), the IVT says that there is some *c* with -2 < *c* < 2 and *f*(*c*) = 0.