Zhang Yu's eight theorems of remembering high numbers refer to: zero theorem, maximum theorem, mean value theorem, Fermat theorem, Rolle theorem, Lagrange mean value theorem, Cauchy mean value theorem and integral mean value theorem.

For example:

1, zero theorem

Let the function f(x) be continuous in the closed interval [a, b], and f(a) and f(b) have different signs (that is, f (a) × f (b).

2. Maximum theorem

If the function f is continuous in the closed interval [a, b], then f has a maximum value and a minimum value in [a, b].

3. Intermediate value theorem

Because f(x) is continuous on [a, b], there is a maximum value m and a minimum value n on [a, b]; That is to say, for all x∈[a, b], there is n.

Therefore, there is n

So n < = [f (x1)+f (x2)+...+f (xn)]/n <; =M, so there is at least a little c in (x 1, xn), so f (c) = [f (x1)+f (x2)+...+f (xn)]/n.

4. Fermat Theorem

The function f(x) is defined in a neighborhood U(ξ) of point ξ, where it is derivable. For any x∈U(ξ), there is f(x)≤f(ξ) (or f(x)≥f(ξ).

5. Rolle theorem

The function f(x) satisfies the following conditions:

(1) is continuous on the closed interval [a, b];

(2) Derivable in (a, b);

(3)f(a)= f(b)；

Then at least one ξ∈(a, b) exists so that f'(ξ)=0.

6. Lagrange mean value theorem

If the function f(x) is derivable on (a, b) and continuous on [a, b], there must be ξ∈(a, b) such that f'(ξ)*(b-a)=f(b)-f(a), f(x).

7. Cauchy mean value theorem

If the functions f(x) and F(x) satisfy:

(1) is continuous on closed intervals a and b;

(2) Derivable in the open interval (a, b);

(3) For any x∈(a, b), F'(x)≠0,

Then there is at least one zeta in (a, b), which makes the equation f (b)-f (a)/f (b)-f (a) = f' (zeta)/f' (zeta) hold.

8. Integral mean value theorem

If the function f(x) is continuous in the closed interval [a, b], there is at least one point ξ in the integral interval [a, b], so the following formula holds.

∫ lower limit a upper limit b f(x)dx=f(ξ)(b-a) (a≤ ξ≤ b).