Essential Concepts

• If [latex]f[/latex] is continuous over [latex][a,b][/latex] and differentiable over [latex](a,b)[/latex] and [latex]f(a)=0=f(b)[/latex], then there exists a point [latex]c \in (a,b)[/latex] such that [latex]f^{\prime}(c)=0[/latex]. This is Rolle’s theorem.
• If [latex]f[/latex] is continuous over [latex][a,b][/latex] and differentiable over [latex](a,b)[/latex], then there exists a point [latex]c \in (a,b)[/latex] such that
  [latex]f^{\prime}(c)=\dfrac{f(b)-f(a)}{b-a}[/latex].

  This is the Mean Value Theorem.

• If [latex]f^{\prime}(x)=0[/latex] over an interval [latex]I[/latex], then [latex]f[/latex] is constant over [latex]I[/latex].
• If two differentiable functions [latex]f[/latex] and [latex]g[/latex] satisfy [latex]f^{\prime}(x)=g^{\prime}(x)[/latex] over [latex]I[/latex], then [latex]f(x)=g(x)+C[/latex] for some constant [latex]C[/latex].
• If [latex]f^{\prime}(x)>0[/latex] over an interval [latex]I[/latex], then [latex]f[/latex] is increasing over [latex]I[/latex]. If [latex]f^{\prime}(x)<0[/latex] over [latex]I[/latex], then [latex]f[/latex] is decreasing over [latex]I[/latex].