document.write( "Question 262269: For what values of x is the function f(x)= 5+15x+6x^2-x^3 decreasing? \n" ); document.write( "

Algebra.Com's Answer #193207 by drk(1908) $\"\"$ $\"About$

You can put this solution on YOUR website!
here is the problem:
\n" ); document.write( "(i) $\"f%28x%29=+5%2B15x%2B6x%5E2-x%5E3\"$
\n" ); document.write( "to determine increasing and decreasing, take a second derivative as
\n" ); document.write( "(ii) $\"f%27%28x%29+=+15+%2B+12x+-+3x%5E2\"$
\n" ); document.write( "set = 0 and find x. We get
\n" ); document.write( "(iii) $\"f%27%28x%29+=+15+%2B+12x+-+3x%5E2+=+0+\"$
\n" ); document.write( "divide by -3 to get
\n" ); document.write( "(iv) $\"x%5E2+-+4x+-+5+=+0\"$
\n" ); document.write( "(v) $\"%28x%2B1%29%28x-5%29+=+0\"$
\n" ); document.write( "so, x = -1 and x = 5.
\n" ); document.write( "these are critical values.
\n" ); document.write( "From (ii) take a second derivative as
\n" ); document.write( "(vi) f''(x) = $\"12+-+6x+=+0\"$
\n" ); document.write( "and solving for x, we get
\n" ); document.write( " $\"x+=+2\"$
\n" ); document.write( "this is also a critical value.
\n" ); document.write( "pick a number less than and greater than 2 in to f'' and we get
\n" ); document.write( "x= 0 - - > increasing ; x = 4 - - > decreasing.
\n" ); document.write( "--
\n" ); document.write( "so the intervals of increasing and decreasing are
\n" ); document.write( "increasing (-00, 2)
\n" ); document.write( "decreasing (2, +00)
\n" ); document.write( "here is the graph to help.\r
\n" ); document.write( "\n" ); document.write( " $\"+graph%28+300%2C+200%2C+-10%2C+10%2C+-20%2C+100%2C+5%2B15x%2B6x%5E2-x%5E3%29+\"$ \n" ); document.write( "

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