document.write( "Question 515247: find the maximum value of the function\r
\n" ); document.write( "\n" ); document.write( "f(x)= 3+4x^2-x^4\r
\n" ); document.write( "\n" ); document.write( "hint: let t=x^2 \n" ); document.write( "

Algebra.Com's Answer #343871 by drcole(72) $\"\"$ $\"About$

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The first thing to notice here is that f(x) is a continuous function, and that, if we took its limit as x approaches $\"%2Binfinity\"$ or $\"-infinity\"$ , the function approaches $\"-infinity\"$ because of the coefficient of the leading term ( $\"x%5E4\"$ ) is -1 and $\"x%5E4\"$ goes to $\"infinity\"$ in either direction. So this function will not have an absolute minimum (it can be as negative as we want), but because it is continuous everywhere and approaches $\"-infinity\"$ in either direction, it WILL have an absolute maximum.\r
\n" ); document.write( "\n" ); document.write( "We also notice that this function has a continuous derivative everywhere, since it is a polynomial. If we know that a function has a continuous derivative everywhere, and we know that it has an absolute maximum, that maximum must be at a stationary point: a point where the first derivative equals zero. So we need to find the derivative, set it equal to zero, and solve for x.\r
\n" ); document.write( "\n" ); document.write( " $\"+f%28x%29+=+3+%2B+4x%5E2+-+x%5E4+\"$
\n" ); document.write( " $\"+%28df%28x%29%29%2F%28dx%29+=+0+%2B+4%282x%29+-+4x%5E3+\"$ (applying the power rule for differentiation)
\n" ); document.write( " $\"+%28df%28x%29%29%2F%28dx%29+=+8x+-+4x%5E3+\"$ (simplifying)
\n" ); document.write( " $\"+0+=+8x+-+4x%5E3+\"$ (setting the first derivative equal to zero)
\n" ); document.write( " $\"+0+=+x%5E3+-+2x+\"$ (dividing both sides by -2)
\n" ); document.write( " $\"+0+=+x%28x%5E2+-+2%29+\"$ (factoring out an x)
\n" ); document.write( " $\"+0+=+x%28x+%2B+sqrt%282%29%29%28x+-+sqrt%282%29%29+\"$ (factoring using the difference of two squares)\r
\n" ); document.write( "\n" ); document.write( "Setting each factor equal to 0, we get the roots $\"x+=+0\"$ , $\"x+=+sqrt%282%29+\"$ , and $\"x+=+-sqrt%282%29\"$ . These are the only possible x-values for the location of the maximum of f(x). We now plug each of these roots into f(x) and find the largest value among the three for the maximum.\r
\n" ); document.write( "\n" ); document.write( " $\"+f%280%29+=+3+%2B+4%280%5E2%29+-+0%5E4+=+3\"$
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\n" ); document.write( "\n" ); document.write( "So the maximum value of f(x) is 7, and it occurs at two values of x: $\"x+=+sqrt%282%29+\"$ and $\"x+=+-sqrt%282%29\"$ . \n" ); document.write( "

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