Question

Graph the following function:

y= e^x / x²

Ensure that you include the following properties:

• Asymptote, x, y – intercepts, Domain and Range
• First derivative, Maximum and Minimum
• Second derivative, inflection points
• Regions or increasing and decreasing
• Regions or concave up and concave down
• Sketch of the graph

Answer 1

Assimptotes x=0, y=0

It is no x,y- interceptes.

$Domain: ( − ∞ , 0 ) ∪ ( 0 , ∞ ) , { x | x ≠ 0 }$

$Range: ( 0 , ∞ ) , { y | y > 0 }$

$y'=\frac{e^x(x-2)}{x^3}$

$\frac{e^x(x-2)}{x^3}=0$ , x=2, $y=e^2/4$

$y"=-\frac{4xe^x-x^2e^x-6e^6}{x^4}$

At x=2 y">0, so

(2,e²/4) -local minimum

No inflection points, because $y" \not=0$

$Increasing\ on: ( − ∞ , 0 ) , ( 2 , ∞ )$

$Decreasing\ on: ( 0 , 2 )$

$Concave\ up\ on ( − ∞ , 0 ) since f '' ( x ) is\ positive$

$Concave\ up\ on ( 0 , ∞ ) since f '' ( x ) is\ positive$