Chapter 3: mean value theorems — Advanced Mathematics - Collection of Formulas

Chapter 3: mean value theorems

3.1 Three major theorems

Rolle's theorem

• Definition:
• If three conditions are met:
• 1)$f$ on $[a, b]$ continuity;
• 2)$f$ is differentiable within $(a, b)$;
• 3)f (a)=f (b)
• So, it can be known: $$\text{ then }\exists\xi\in(a,b)\text{ , to use }f^{\prime}(\xi)=0$$
• diagram:
• 

Lagrange's mean value theorem

• Definition:
• If the following conditions are met:
• 1)$f$ on $[a, b]$ continuity
• 2)$f$ is differentiable within ($a, b)$
• Therefore, there exists $\xi\in(a,b)$ such that $$f(b)-f(a)=f^{\prime}(\xi)(b-a).$$
• diagram:
• 

Cauchy's mean value theorem

• Definition:
• If the following conditions are met:
• 1) $f, F$ on $[a, b]$ continuity $; $
• 2) $f, F$ is differentiable within $(a, b)$, and $\forall x\in(a, b), F^{\prime}(x)\neq0$
• Therefore, there exists $\xi\in(a,b)$ such that $$\frac{f(b)-f(a)}{F(b)-F(a)}=\frac{f^{\prime}(\xi)}{F^{\prime}(\xi)}.$$
• diagram:
• 

Relationship of three major theorems

3.2 derivative application

Convexity

• Definition:
• Let the function $f(x)$ be continuity on the interval $I$, if for any two points on $I x_1, x_2$
• 1. If $f\left(\frac{x_1+x_2}{2}\right)<\frac{f(x_1)+f(x_2)}{2}$ for every $x_1,x_2\in I$, then $f$ is strictly convex on $I$.
• 2. If $f\left(\frac{x_1+x_2}{2}\right)>\frac{f(x_1)+f(x_2)}{2}$ for every $x_1,x_2\in I$, then $f$ is strictly concave on $I$.
• Method:
• Use the positive and negative values of the first derivative -> determine the monotonicity of the function;
• Use the positive and negative values of the second derivative -> determine the concavity of the function;
• Test:
• Suppose $f$ is continuous on $[a,b]$ and twice differentiable on $(a,b)$.
• 1) If $f''(x)>0$ on $(a,b)$, then $f$ is convex (concave upward).
• 2) If $f''(x)<0$ on $(a,b)$, then $f$ is concave (concave downward).

Inflection Point

• Definition:
• A point on a continuous curve where its concavity changes.
• Possible value points:
• 1. ${f^{\prime\prime}(x_0)=0}$；
• 2. Points where the second derivative does not exist;

Asymptotes

• Definition:
• 1) Horizontal asymptote: if $\lim_{x\to+\infty}f(x)=A$ or $\lim_{x\to-\infty}f(x)=A$, then $y=A$ is a horizontal asymptote in the corresponding direction.
• 2) Vertical asymptote: if a one-sided limit of $f(x)$ at $x_0$ is $\pm\infty$, then $x=x_0$ is a vertical asymptote.
• 3) Oblique asymptote: if $a=\lim_{x\to\pm\infty}\frac{f(x)}x$ and $b=\lim_{x\to\pm\infty}(f(x)-ax)$ are finite, then $y=ax+b$ is an oblique asymptote in that direction.
• Method:
• 1. Determine the horizontal asymptote -> When x -> is infinite, y tends toward a finite value;
• 2. Determine the vertical asymptote -> When x -> a certain point (finite value), y approaches infinity;
• 3. Determine the oblique asymptote
• -> $\lim_{x\to\infty}\frac{f(x)}x=a$ Finite value;
• -> $\lim_{x\to\infty}(f(x)-ax)=b$ exist
• At this time, there is an oblique asymptote;