## Analysis I: Lecture 12

In which we study some implications of Rolle’s theorem and the Mean value theorem.

• Theorem 26 (Inverse function theorem) Let $f:[a,b] \to \mathbb{R}$ be continuous on $[a,b]$ and differentiable on $(a,b)$ with $f'(x) > 0$ for all $x \in (a,b)$.  Let $c = f(a)$ and $d = f(b)$.  Then the function $f : [a,b] \to [c,d]$ is a bijection, and $f^{-1} : [c,d] \to [a,b]$ is continuous on $[c,d]$ and differentiable on $(c,d)$ with $\displaystyle (f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}$ for all $y \in (c,d)$.  We used Corollary 25(i) to show that the function is strictly increasing, and then used Theorem 20 to check that the function is a bijection.  We then checked that the inverse is differentiable by going back to our definition of differentiability.
• Theorem 27 (Cauchy‘s mean value theorem) Let $f$ and $g$ be functions from $[a,b]$ to $\mathbb{R}$ that are continuous on $[a,b]$ and differentiable on $(a,b)$.  Then there is some $c \in (a,b)$ such that $[f(b) - f(a)] g'(c) = [g(b) - g(a)] f'(c)$.  We noted that if $g'(x) \neq 0$ for all $x \in (a,b)$, then this tells us that there is some $c \in (a,b)$ such that $\displaystyle \frac{f(b) - f(a)}{g(b) - g(a)} = \frac{f'(c)}{g'(c)}$.  To prove the result, we applied Rolle’s theorem to a suitable auxiliary function.
• Definition of functions that are continuously differentiable or twice differentiable or $n$ times differentiable.
• Theorem 28 (Taylor‘s theorem with Lagrange‘s form of the remainder) Let $f : [a,a+h] \to \mathbb{R}$ be a function such that $f$ and its first $n-1$ derivatives are continuous on $[a,a+h]$ and $f$ is $n$ times differentiable on $(a,a+h)$.  Then there exists $\theta \in (0,1)$ such that $\displaystyle f(a+h) = f(a) + h f'(a) + \frac{h^2}{2} f''(a) + \dotsb + \frac{h^{n-1}}{(n-1)!} f^{(n-1)}(a) + \frac{h^n}{n!} f^{(n)} (a + \theta h)$. We shall prove this next time.

#### Understanding today’s lecture

You could experiment with the inverse function theorem on some more examples, to get a feel for what it says.

Is the inverse function theorem compatible with the chain rule?  For suitable functions $f$, we have $f^{-1}(f(x)) = x$ — what happens if we look at the derivative of each side?

Are you comfortable with why the two versions of Cauchy’s mean value theorem are equivalent if $g'(x) \neq 0$ for all $x \in (a,b)$?

You could add the various new properties of functions that we met today to your functions grid.  Can you find a function that is differentiable but not continuously differentiable?  A function that is continuously differentiable but not differentiable?  A function that is continuously differentiable but not twice differentiable?  A function that is twice differentiable but not continuously differentiable?  And so on!  This kind of exercise is really, really good for getting a proper understanding of the definitions.

Lots of mathematicians’ names today, so several biographies for you to read above.  There’s loads of information about Taylor’s theorem on Wikipedia (and in introductory analysis books, of course).  If you are so inclined, you could get a computer to draw pretty pictures.

#### Preparation for Lecture 13

Can you prove Taylor’s theorem in the form we stated it today?  You might like to start with $n = 1$, $n = 2$, etc.