Help with the graph of a derivative, picture attached.

Question

juscallmesteve · Answer

attachment

solomonzelman · Answer

** The rule is ** [1] When $\color{#000000}{ \displaystyle f'(x)>0 }$, the $\color{#000000}{ \displaystyle f(x) }$ increases. [2] When $\color{#000000}{ \displaystyle f'(x)<0 }$, the $\color{#000000}{ \displaystyle f(x) }$ decreases. $\color{#000000}{ \LARGE ^\text{_______________________________________} }$ This should make sense, if you get the following definitions: [a] $\color{#000000}{ \displaystyle f'(x) }$, the derivative of the function $\color{#000000}{ \displaystyle f(x) }$. [b] $\color{#000000}{ \displaystyle f'(c) }$, is the $\color{#000000}{ \displaystyle f'(x) }$ evaluated at $\color{#000000}{ x=c }$, or the instantaneous slope of the function $\color{#000000}{ \displaystyle f(x) }$, at $\color{#000000}{ x=c }$. [c] $\color{#000000}{ \displaystyle f(x) }$ is going up when the slope is positive. For this reason, $\color{#000000}{ \displaystyle f(x) }$ is said to be increasing when $\color{#000000}{ \displaystyle f'(x)>0 }$. [d] $\color{#000000}{ \displaystyle f(x) }$ is going down when the slope is positive. For this reason, $\color{#000000}{ \displaystyle f(x) }$ is said to be decreasing when $\color{#000000}{ \displaystyle f'(x)<0 }$.