Math Natural exponential function Curve sketching

Curve sketching of the natural exponential function

Even with natural exponential functions a curve sketching can be done



When setting to zero and calculating an equation with $e$, keep in mind that $e$ to the power of anything never becomes zero.

$e^{x}>0$ with $x\in\mathbb{R}$


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

    $x\cdot e^x=0$

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

    $e^x>0$ (can never be zero!) and
  3. Extrema

    Calculate extrema: Set first derivative equal to zero

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

    use suspicious points for extrema in the second derivative test:

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

    Calculate the y-coordinate and specify the minimum:

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

  4. Inflection point

    Calculate inflection point: Set second derivative equal to zero

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

    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:

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