### Differentiation: Applications of derivatives

### Types of increasing and decreasing

We can further refine the increase and decrease of functions. For this we use the second derivative.

If #f'(x)>0# and #f''(x)>0#, then #f# is #\blue{\text{concave up and increasing }}#.

The function #f(x)=x^3# is concave up and increasing for #x>0#, because \[\begin{array}{rcl}f'(x)&=&3x^2\;>\;0\\f''(x)&=&6x\;>\;0 \end{array}\]

If #f'(x)>0# and #f''(x)<0#, then #f# is #\green{\text{concave down and increasing}}#.

The function #f(x)=x^3# is concave down and increasing for #x<0#, because \[\begin{array}{rcl}f'(x)&=&3x^2\;>\;0\\f''(x)&=&6x\;<\;0\end{array}\]

If #f'(x)<0# and #f''(x)\lt 0#, then #f# is #\orange{\text{concave down and decreasing}}#.

The function #f(x)=-x^5# is concave down and decreasing for #x>0#, because \begin{array}{rcl}f'(x)&=&-5x^4\;<\;0\\f''(x)&=&-20x^3\;\lt\;0\end{array}

If #f'(x)<0# and #f''(x)\gt 0#, then #f# is #\purple{\text{concave up and decreasing}}#.

The function #f(x)=-x^5# is concave up and increasing for #x\lt 0#, because \begin{array}{rcl}f'(x)&=&-5x^4\;<\;0\\f''(x)&=&-20x^3\;\gt\;0\end{array}

Here we see the function \[f(x)=\sin(\frac{1}{2}\pi\cdot x)+1\] This function is

- #\green{\text{concave down and increasing}}\ (dashed)# between #0# and #1#
- #\orange{\text{concave down and decreasing}}\ (dotted)# between #1# and #2#
- #\purple{\text{concave up and decreasing}}\ (solid)# between #2# and #3#
- #\blue{\text{concave up and increasing}}\ (long\_short\ dashed)# between #3# and #4#

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