### Graph of a Quartic Function with zeros(roots) at [-20, -20, -15, -2]

 Let's analyze the following polynomial function.f(x) = (x+15)(x+20)2      = x432You might also be interested in reading about quadraticcubic functions Let us analyze the graph of this function which is a quartic polynomial. A quartic polynomial is a fourth degree polynomial.  These have zero to four real rootsThey may have one, two or three extrema (maxima/minima - turning points)Zero, one or two inflection (inflexion) points.Let's do some curve sketchingy = f(x) = (x+15)(x+20)2= x432We can plot the curve by computing some sample points other than the zeroes already known. Some sample points are: A few computed points on the curve, apart from the zero(s) which are known:Let us inspect the roots of the given polynomial function.These values of x are the roots of the quadratic equation (x+15)(x+20)2Let us analyze the turning points in this curve. The derivative of the given function = f'(x) = 4x32This is a cubic function. Turning points are those where the derivative of the function (obtained on differentiating the function) is zero. These may be obtained by solving the cubic equation 4x32Some of these are local maximas and some are local minimasThe turning points of this curve are approximately at x = [-20.0, -16.7, -5.9]. At these points, the curve has either a local maxima or minima. These are the extrema - the peaks and troughs in the graph plot..Now, let us take a look at the inflexion These are the points where the curve changes concavity.  The second derivative f'''(x) = 12x32This is a quadratic function. The value(s) of x for which this quadratic function is zero, are the inflexion points. This curve has two inflexion points at (approx) -18.5 and -10.0. At these points, the concavity changes.Check the plot of another quartic curve hereHere's another quartic curve hereHere's another quartic curve hereIIT JEE syllabusCommon CoreGCSE
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Prashant Bhattacharji,
Mar 20, 2017, 12:26 AM