document.write( "Question 534950: At what value(s) of x on the curve y=−9+250x 3 −3x 5 does the tangent line have the largest slope? \n" ); document.write( "
Algebra.Com's Answer #351762 by KMST(5328)\"\" \"About 
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If that's really
\n" ); document.write( "\"y=9%2B250x%5E3-3x%5E5\"
\n" ); document.write( "(it is not clear, the way it shows onscreen),
\n" ); document.write( "it is a polynomial. It is unconventionally written with the terms in order of increasing degree. (It's usually done in reverse order).
\n" ); document.write( "Because of the negative coefficient on the \"leading\" term \"%28-3x%5E5%29\"
\n" ); document.write( "the polynomial decreases with x for very large values of x.
\n" ); document.write( "Because its degree is odd (5), it also decreases with x for negative x that are very large in absolute value.
\n" ); document.write( "It comes from y=infinity at x=-infinity and goes to y=-infinity at infintely positive x. The absolute values of the slope, and of the polynomial see no limit as the absolute value of x increases, but the slope is negative at both ends.
\n" ); document.write( "To see what it does in the middle, I would calculate first and second derivatives.
\n" ); document.write( "First derivative; \"750x%5E2-15x%5E4\"
\n" ); document.write( "Second derivative; \"1500x-60x%5E3\"
\n" ); document.write( "The first derivative is really the slope of the tangent. We knew that the function decreased (negative slope of the tangent) at both ends. Now we will know just how far those ends extend, and what happens in the middle.
\n" ); document.write( "The first derivative is an even function, symmetrical with respect to the y-axis, with the same value for \"x\" and \"-x\".
\n" ); document.write( "It is negative for \"x%3C-sqrt%2850%29\" and \"x%3Esqrt%2850%29\"
\n" ); document.write( "indicating negative slope, decreasing function.
\n" ); document.write( "It is zero at \"x=-sqrt%2850%29\", \"x=0\", \"x=sqrt%2850%29\"
\n" ); document.write( "indicating horizontal tangents with zero slope.
\n" ); document.write( "It does not change sign at zero, but stays positive on both sides for \"-sqrt%2850%29%3Cx%3C0\" and \"0%3Cx%3Csqrt%2850%29\".
\n" ); document.write( "The first derivative (slope) does change sign from negative to positive at \"x=-sqrt%2850%29\"
\n" ); document.write( "and from positive to negative at \"x=sqrt%2850%29\".
\n" ); document.write( "For the function, it means that the function has a local minimum at \"x=-sqrt%2850%29\",
\n" ); document.write( "a saddle point (inflection with horizontal tangent) at \"x=0\",
\n" ); document.write( "and a local maximum at \"x=sqrt%2850%29\".\r
\n" ); document.write( "\n" ); document.write( "Between the local minimum and local maximum, the function just continuously increases, but the slope of the tangent goes through many changes in that interval.
\n" ); document.write( "We knew that the slope of the tangent was negative at both ends.
\n" ); document.write( "In the middle, it increases (goes from zero to positive, decreases back to zero, increases once again, and goes back to zero. At the two places where it is positive it goes through a local maximum (maximum for that interval). Because of its symmetry, the value of the slope at both of those maxima is the same , and because the slope is negative elsewhere, that value is the absolute maximum value for the slope.
\n" ); document.write( "The second derivative tells you where the maxima and minima of the first derivative are. (Those are the inflection points of the function).
\n" ); document.write( "The second derivative is zero and changes sign (from negative to positive) at \"x=0\"
\n" ); document.write( "where the first derivative (the slope of the tangent) goes trough a local minimum of zero.
\n" ); document.write( "The second derivative is also zero at \"x=-5\", and \"x=5\".
\n" ); document.write( "It goes from positive to negative at those points indicating that the local maxima, that we knew are the absolute maxima, of the first derivative and slope occur at those points (greatest slope of the tangent).
\n" ); document.write( " The graph shows the function in red and the first derivative (the slope of the tangent) in green. \n" ); document.write( "
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