Fermat’s Theorem(Stationary Points)

• If $f^{\prime }(c)$ exists
• If $f(x)$ has a local/global maximum/minimum at $x=c$, then: $f^{\prime }(c)$ = 0 In other words, local max/min $⟹$ $f^{\prime }(c)=0$

Inverse Logic Does not apply

$f^{\prime }(0)$ does not necessarily mean that there is a local minima or maxima. Take $f(x)=x^3$, which has a $f^{\prime }(0)$ at 0, but is not a local minima or maxima.

Proof

• Fermat’s Theorem Stationary Points Proof