(a) Let f(x)=x^2 and g(x)=-x^3+x^2+3 x+2 . Then f(-1)=g(-1)...

(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 that 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 fat $(c, f(c))$ is parallel to the tangent line to $g$ at $(c, g(c)).$

(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 that 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 fat $(c, f(c))$ is parallel to the tangent line to $g$ at $(c, g(c)).$

Key Concepts

Tangent Lines and Parallelism

A tangent line to a function at a given point is a straight line that just touches the curve at that point and has the same slope as the function at that point. Two tangent lines are parallel if and only if their slopes are equal. In this problem, finding a point where the tangent lines to both functions f and g are parallel is equivalent to finding a point c where the derivatives (slopes) f?(c) and g?(c) are equal.

Differentiability

Differentiability of a function implies that the function has a well-defined tangent line at every point in its interval. This property is crucial because it ensures that the derivative, which represents the slope of the tangent line, exists at every point within the interval of interest. In the context of this problem, both functions f and g are differentiable, so their tangent lines are meaningful and can be compared by their slopes.

Rolle's Theorem

Rolle's Theorem is a specific case of the Mean Value Theorem that applies when a continuous and differentiable function takes equal values at the endpoints of an interval. It guarantees the existence of at least one point c within the interval where the first derivative of the function is zero. In the problem, by considering the function h(x) = g(x) - f(x), and knowing that h(-1)=h(2) in part (a) or h(a)=h(b) in part (b), Rolle's Theorem can be used to conclude that there is at least one point c in the interval where h?(c) = 0, which is equivalent to f?(c) = g?(c).

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