document.write( "Question 1155435: Locate the critical points of the following function. Then use the second derivative Test to determine whether they correspond to local maxima, local minima, or neither.
\n" ); document.write( "f(x)=-e^x (x-9) \n" ); document.write( "

Algebra.Com's Answer #778031 by greenestamps(13215) $\"\"$ $\"About$

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\n" ); document.write( " $\"f%28x%29+=+-e%5Ex%28x-9%29+=+%289-x%29e%5Ex\"$

\n" ); document.write( "Use the product rule to find the derivative:
\n" ); document.write( " $\"df%2Fdx+=+%289-x%29e%5Ex-1%28e%5Ex%29+=+%288-x%29e%5Ex\"$

\n" ); document.write( "The derivative is zero only at x=8; the function value at x=8 is $\"%289-8%29e%5E8+=+e%5E8\"$

\n" ); document.write( "The critical point is (8,e^8).

\n" ); document.write( "Use the product rule to find the second derivative:
\n" ); document.write( " $\"d%5E2f%2Fdx%5E2+=+%288-x%29e%5Ex-1%28e%5Ex%29+=+%287-x%29e%5Ex\"$

\n" ); document.write( "The value of the second derivative at x=8 is $\"%287-8%29e%5E8+=+-e%5E8\"$

\n" ); document.write( "The second derivative is negative at x=8; the critical point is a maximum.

\n" ); document.write( " $\"graph%28400%2C400%2C-2%2C12%2C-1000%2C5000%2C%289-x%29e%5Ex%29\"$
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