Let $f\left(x\right)$ be continuous on $[𝒶,𝒷]$ and differentiable on $(𝒶,𝒷)$

• if $\forall x\in (𝒶,𝒷),f\prime \left(x\right)>0$ then $f\left(x\right)$ is increasing on $[𝒶,𝒷]$
• if $\forall x\in (𝒶,𝒷),f\prime \left(x\right)<0$ then $f\left(x\right)$ is decreasing on $[𝒶,𝒷]$

Let $\omega$ be a critical point then

• if $f\prime \left(x\right)$ changes from positive to negative at $\omega$ then $f\left(x\right)$ has a local maximum at $\omega$
• if $f\prime \left(x\right)$ changes from negative to positive at $\omega$ then $f\left(x\right)$ has a local minimum at $\omega$
• if $f\prime \left(x\right)$ does not change sign, then it has neither a local minimum nor a local maximum at $\omega$

Let $f\left(x\right)$ be twice differentiable on an open interval $ℐ$, then

• if $f^{″}>0$ on $ℐ$, then $f\left(x\right)$ is concave up on $ℐ$;
• if $f^{″}<0$ on $ℐ$, then $f\left(x\right)$ is concave down on $ℐ$;

Suppose $f^{\prime }\left(\omega \right)=0$ and $f^{″}\left(x\right)$ is continuous on an open interval containing $\omega$ then

• if $f^{″}\left(\omega \right)<0$ then $\omega$ is a local maximum;
• if $f^{″}\left(\omega \right)>0$ then $\omega$ is a local minimum;
• if $f^{″}\left(\omega \right)=0$ then the test is inconclusive.