document.write( "Question 4122: Estimate the intervals on which the function g(x)=x^3-x^2+2x-8 is increasing and decreasing \n" ); document.write( "

Algebra.Com's Answer #1814 by ivankst(3) $\"\"$ $\"About$

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This type of problems is solved by finding the derivative of the function and estimating intervals on which the derivative is positive or negative. When derivative is positive, function is increasing and when derivative is negative, function is decreasing.
\n" ); document.write( "Let's first find the derivative:
\n" ); document.write( " $\"dg%2Fdx=3x%5E2-2x%2B2\"$
\n" ); document.write( "Now let's find values of x for which $\"dg%2Fdx\"$ =0 :
\n" ); document.write( " $\"3x%5E2-2x%2B2=0\"$
\n" ); document.write( "This is solved as every quadratic equation:
\n" ); document.write( " $\"x=%282%2B-sqrt%284-4%2A3%2A2%29%29%2F6\"$
\n" ); document.write( "Now notice the value in square root: $\"4-4%2A3%2A2=4-24=-20\"$
\n" ); document.write( "It's negative!!! That means that this quadratic equation doesn't have solutions in field of real numbers which means that $\"dg%2Fdx\"$ doesn't have x-intercepts : it's either all positive or all negative.
\n" ); document.write( "You can see that its all positive in 2 ways:
\n" ); document.write( "1) coefficient of x^2 is 3 which is a positive number, so it's all positive.
\n" ); document.write( "2) for example find $\"dg%2Fdx\"$ (0). It's obvious that $\"dg%2Fdx\"$ (0)=2. 2 is a positive number so the function is all positive.
\n" ); document.write( "Now since $\"dg%2Fdx\"$ is always positive, function g(x) is ALWAYS increasing.
\n" ); document.write( "Matematically correct: g(x) is increasing for x in whole real numbers field.
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