Find the coordinates of points where the graph of f ( x ) = x 4 − 4 x 3 + 8has horizontal tangents. As a check, graph f(x) and see whether the points you found look as though they have horizontal tangents.

Question

Find the coordinates of points where the graph of  f  (  x  )  =      x    4    −  4      x    3    +  8has horizontal tangents. As a check, graph f(x) and see whether the points you found look as though they have horizontal tangents.

alinearjb · Accepted Answer

The function is   f  (  x  )  =      x    4    −  4      x    3    +  8The tangent to a curve is horizontal i. e parallel to x axis slope of curveis zero i.e             d      y              d      x        =      f    ′    (  x  )  =  0Now       f    ′    (  x  )  =  4      x    3    −  12      x    2    +  0So       f    ′    (  x  )  =  0  ⇒  4      x    3    −  12      x    2    =  0  ⇒      x    2    (  4  x  −  12  )  =  0    ⇒  x  =  0  ⇒  4  x  −  12  =  0    ⇒  x  =  3Hence at x=0 and x=3 the tangent to the curve is horizontal.Now at x=0,   f  (  0  )  =      0    4    −  4  ⋅      0    3    +  8i.e   (  x  ,  y  )  =  (  0  ,  8  )Also at   x  =  3  ,     f  (  3  )  =      3    4    −  4  ⋅  3  +  8    =  81  −  108  +  8    =  89  −  108  =  −  19i.e   (  x  ,  y  )  =  (  3  ,  −  19  )

Find the coordinates of points where the graph of f(x) = x^4 − 4x^3 + 8 has horizontal tangents. As a check, graph f(x) and see whether the points you found look as though they have horizontal tangents.

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