Curve sketching of the natural exponential function

Even with natural exponential functions a curve sketching can be done

Example

Examine $f(x)=x\cdot e^x$ on the following properties:

• Zeros
• Extrema
• Inflection point

1. Calculate derivative

   To compute the drivative, use the product rule.
   

   $f(x)=x\cdot e^x$

   $f'(x)=x\cdot e^x+e^x$ $=e^x(x+1)$
   

   $f''(x)=x\cdot e^x+e^x+e^x$ $=e^x(x+2)$
   

   $f'''(x)=x\cdot e^x+e^x+e^x+e^x$ $=e^x(x+3)$

2. Zeros

   Calculating Zeros: Set function equal to zero
   

   $f(x)=0$
   $x\cdot e^x=0$

   Product of zero: A product becomes zero if one of the factors becomes zero.

   $e^x>0$ (can never be zero!) and
   $x_N=0$

3. Extrema

   Calculate extrema: Set first derivative equal to zero
   $f'(x)=0$
   $e^x(x+1)=0$

   $e^x>0$ (can never be zero!) and
   $x+1=0\quad|-1$
   $x_E=-1$

   use suspicious points for extrema in the second derivative test:

   $f''(-1)=e^{-1}>0$ => minimum
   
   

   Calculate the y-coordinate and specify the minimum:

   $f(-1)$ $=-1\cdot e^{-1}$ $=-e^{-1}$ $\approx-0.37$
   
   $T(-1|-0.37)$

4. Inflection point

   Calculate inflection point: Set second derivative equal to zero
   $f''(x)=0$
   $e^x(x+2)=0$
   

   $e^x>0$ (can never be zero!) and
   $x+2=0\quad|-2$
   $x_W=-2$

   Insert point suspected of having an inflection point in the third derivative:

   $f'''(-2)=e^{-2}\neq0$ => Inflection point
   
   

   Calculate y-coordinate and specify inflection point:

   $f(-2)$ $=-2e^{-2}$ $\approx-0.27$
   
   $W(-2|-0.27)$