document.write( "Question 1155432: Perform a first derivative test on the function f(x)= x sqrt(100-x^2);[-10,10].
\n" ); document.write( "a. Locate the critical points of the given function.
\n" ); document.write( "b. Use the first derivative test to locate the local maximum and minimum values.
\n" ); document.write( "c.identify the absolute minimum and maximum values of the function on the given intervals(when they exist) \n" ); document.write( "
Algebra.Com's Answer #778032 by greenestamps(13200)\"\" \"About 
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\n" ); document.write( "\"f%28x%29+=+x%2Asqrt%28100-x%5E2%29+=+x%28100-x%5E2%29%5E%281%2F2%29\"

\n" ); document.write( "We can note before we start that the function is odd -- that is, f(-x) = -f(x). We can use that, if we find it convenient, to simplify solving the problem.

\n" ); document.write( "Use the product rule to find the derivative.

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\n" ); document.write( "
\n" ); document.write( "\"df%2Fdx+=+%28100-2x%5E2%29%2F%28100-x%5E2%29%5E%281%2F2%29\"

\n" ); document.write( "The derivative is zero when the numerator is zero:

\n" ); document.write( "\"100-2x%5E2=0\"
\n" ); document.write( "\"100+=+2x%5E2\"
\n" ); document.write( "\"x%5E2+=+50\"
\n" ); document.write( "\"x+=+5sqrt%282%29\" or \"x+=+-5sqrt%282%29\"

\n" ); document.write( "Find the function value at the critical point with the positive x value.

\n" ); document.write( "\"f%285sqrt%282%29%29+=+5sqrt%282%29%2Asqrt%28%28100-50%29%29+=+50\"

\n" ); document.write( "The maximum value of the function is at (5*sqrt(2),50); since the function is an odd function, the minimum value is at -5*sqrt(2),-50).

\n" ); document.write( "\"graph%28400%2C400%2C-12%2C12%2C-80%2C80%2Cx%2Asqrt%28100-x%5E2%29%29\"

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