Verified answer

Use Rolle's Theorem on the function h(x) = f´(x) - g´(x).

Since f´(x) and g`(x) are continuous on the closed interval [a, b], the same is true for h(x).

Since f´´(x) and g´´(x) exist on the open interval (a,b), the same is true of h´(x) = f´´(x) - g´´(x).

Finally, since f´(a) = g´(a) and f´(b) = g´(b), h(a) = f´(a) - g´(a) = 0 = f´(b) - g´(b) = h(b).

So h satisfies all conditions of Rolle's Theorem, and therefore there exists a number c in (a,b) such that 0 = h´(c) = f´´(c) - g´´(c), or equivalently f´´(c) = g´´(c).

Lord bless you today!