Using Rolles Theorem (a) Let and Then and Show that there is at least one value in the

Chapter 4, Problem 88

(choose chapter or problem)

Using Rolle’s Theorem

(a) Let \(f(x)=x^{2}\) and \(g(x)=-x^{3}+x^{2}+3 x+2\). Then f(-1) = g(-1) and f(2) = g(2). Show there is at least one value c in the interval (-1, 2) where the tangent line to f at (c, f(c)) is parallel to the tangent line to g at (c,g(c)). Identify c.

(b) Let f and g be differentiable functions on [a, b] where f(a) = g(a) and f(b) = g(b). Show that there is at least one value c in the interval (a, b) where the tangent line to f at (c,f(c)) is parallel to the tangent line to g at (c, g(c)).

Text Transcription:

f(x)=x^2

g(x)=-x^3+x^2+3x+2