Category: Theorems of Differential Calculus

Fermat's Theorem

Knowing derivative of function ${f{{\left({x}\right)}}}$ allows us to make some conclusions about behavior of function ${y}={f{{\left({x}\right)}}}$.

Fermat's Theorem. Suppose function $y={f{{\left({x}\right)}}}$ is defined on interval ${\left[{a},{b}\right]}$ and at some point ${c}\in{\left({a},{b}\right)}$ takes maximum (minimum) value. If exist finite derivative ${f{'}}{\left({c}\right)}$ then ${f{'}}{\left({c}\right)}={0}$.

Rolle's Theorem

Rolle's Theorem. Suppose following three condition hold for function $y={f{{\left({x}\right)}}}$:

1. function is defined and continuous on closed interval ${\left[{a},{b}\right]}$;
2. exists finite derivative ${f{'}}{\left({x}\right)}$ on interval ${\left({a},{b}\right)}$;
3. ${f{{\left({a}\right)}}}={f{{\left({b}\right)}}}$.

then there exists point ${c}$ $\left({a}<{c}<{b}\right)$ such that ${f{'}}{\left({c}\right)}={0}$.

Mean Value Theorem

Mean Value Theorem (Lagrange Theorem). Suppose that function $y={f{{\left({x}\right)}}}$ is defined and continuous on closed interval ${\left[{a},{b}\right]}$ and exists finite derivative ${f{'}}{\left({x}\right)}$ on interval ${\left({a},{b}\right)}$. Then there exists a point $c$ $\left({a}<{c}<{b}\right)$ such that $\frac{{{f{{\left({b}\right)}}}-{f{{\left({a}\right)}}}}}{{{b}-{a}}}={f{'}}{\left({c}\right)}$.