How do you find the points where the graph of the function #f(x)= 3x^5 - 5x^3 + 2# has horizontal tangents and what is the equation?

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Answer 1 · 2016-01-13T02:25:38

#x=-1,0,1#
#(-1,4),(0,2),(1,0)#

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Horizontal tangents occur when the derivative of a function equals #0#.

Find #f'(x)#:

#f(x)= 3x^5 - 5x^3 + 2#
#f'(x)=15x^4-15x^2#

Set the derivative equal to #0#.

#15x^4-15x^2=0#
#15x^2(x^2-1)=0#
#15x^2(x+1)(x-1)=0#

Thus,

#x=0color(white)(ssss)"or"color(white)(ssss)x=-1color(white)(ssss)"or"color(white)(sss)x=1#

We can check a graph:

graph{3x^5-5x^3+2 [-11.82, 13.5, -4.51, 8.15]}