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 $$$\frac{{{f{{\left({b}\right)}}}-{f{{\left({a}\right)}}}}}{{{b}-{a}}}={f{'}}{\left({c}\right)}$$$.