Stationary points

To find the stationary or turning points of a curve  
\[y=f(x)\]
, solve  
\[\frac{d(f(x))}{dx}=0\]
.
To classify them, find  
\[\frac{d^2 (f(x))}{dx^2}\]
. If  
\[\frac{d^2 (f(x))}{dx^2} \gt 0\]
  at the point, the point is a minimum and if  
\[\frac{d^2 (f(x))}{dx^2} \lt 0\]
  at the point, the point is a maximum. If  
\[\frac{d^2 (f(x))}{dx^2}=0\]
, the the point is a stationary point of inflexion.
Example:  
\[f(x)=3x^5-5x^3\]
.
\[\frac{d(f(x))}{dx}15x^4-15x^2=0 \rightarrow 15x^2(x^2-1)=0 \rightarrow x=-1, \; 0, \; 1\]
.
\[\frac{d^2 (f(x))}{dx^2}=60x^3-30x\]

At  
\[x=-1\]
,  
\[\frac{d^2(f(x))}{dx^2}=-30 \lt 0\]
  so this point is a maximum.
At  
\[x=0\]
,  
\[\frac{d^2(f(x))}{dx^2}= 0\]
  so this point is a stationary point of inflexion.
At  
\[x=1\]
,  
\[\frac{d^2(f(x))}{dx^2}=30 \gt 0\]
  so this point is a minimum.

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