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

Key Concepts

Constructing a Difference Function

A common strategy to compare the rates of change of two functions is to define a new function as the difference between them. If the two original functions have the same values at the endpoints of an interval, then their difference function will have identical endpoint values. This construction allows one to apply Rolle’s Theorem to conclude that there is at least one point where the derivative of the difference function is zero, which implies that the derivatives (or slopes of the tangent lines) of the original functions are equal at that point.

Differentiability and Continuity

Both differentiability and continuity are critical prerequisites for applying Rolle’s Theorem and many other fundamental results in calculus. Continuity on a closed interval ensures that a function does not have any abrupt jumps or breaks, while differentiability on the corresponding open interval guarantees that the function has a well-defined tangent at every point in that interval.

Rolle's Theorem

Rolle’s Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and if the function’s values at the endpoints are equal, then there must exist at least one point c in (a, b) where its derivative equals zero. This theorem is a pivotal tool in calculus for establishing the existence of stationary points under appropriate conditions.

Transcript

00:01 This is a well i wouldn't say call it tricky but you kind of figure out what how to proceed with this so they they give us these two functions here x squared and minus x q plus x squared plus plus two and they want us to show prove that there is at least one let's see here let's show that there is at least one value in the interval for minus two minus 1 to 2, says that the tangent line to f at c of x is parallel to the tangent line of c.

00:48 Okay, let's see here.

00:51 Now, what i did here is i kind of did this all in one, both parts of this, because the second part just generalizes the first.

01:03 So what we can do is in the first part, they actually give us this point 1, minus 1 and 2.

01:12 And we can see that we plug in minus 1 here, we get 1, same here, plug in 2, we get 4, and we also get 4 here.

01:20 So what we can do is to find this other function as the difference between these functions, because that means that the difference between these functions, since these functions equal each other at those points, the difference is 0 at those points.

01:33 So we define h.

01:35 So here's actually g and here's f.

01:39 And you can see that they intersect right here and here.

01:43 And we can take the difference of them and we get this h function and we can see where they intersect.

01:48 H is zero, of course.

01:51 So we know that h at minus 1 is 0 and h2 is 0.

01:57 So that means we can use rolls there.

01:59 To say that there has to be at least one point in between minus 1 and 2, such that the derivative of h is 0.

02:10 But the derivative of h is just derivative of f minus the derivative of c.

02:16 So that means the derivative of f equals must equal the derivative of c for some c in this range.

02:23 So that means that, again, that the slope of the tangent line, basically we're saying that the derivative is the same at some.

02:30 Point in this region here.

02:32 So some point between here on here we have the derivative being the same for both functions...

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