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Mathematics-Online problems:

Interactive Problem 161: Minimal Area of a Triangle Constructed by Tangent and Normal Lines of a Power Function

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The shaded triangle is bounded by the $ y-$axis as well as the tangent and normal lines of the graph of the function $ y = x^4$ at the point $ P.$

\includegraphics[width=0.5\linewidth]{TdM_09_A1_bild1_en.eps}

Determine $ y_1$ and $ y_2$ in terms of the $ x-$coordinate of $ P.$ For which $ x_{\min} > 0$ is the area $ A$ of the triangle minimal, and what is the minimum value $ A_{\min}?$


Answer:

$ y_1 = $ $ x^4$    
$ y_2 = $ $ x^4$ + $ /x^2$    
$ x_{\min} = $    
$ A_{\min} = $    

(The results should be correct to four decimal places.)


   

(From: Mathematics Contest 2009)
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  automatically generated: 2/ 6/2018