Find the critical points and the intervals on which the function(x) = 8x^5 - 7x^

Question

Find the critical points and the intervals on which the function(x) = 8x^5 - 7x^3 + 1 is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local minimum or maximum (or neither). (Use symbolic notation and fractions where needed. Give your answer in the form of comma separated list. Enter NONE if there are no critical points.) The critical points with local minimum: The critical points with local maximum: (Use symbolic notation and fractions where needed. Give your answers as intervals in the form (*,*). Use mf for infinity, U for combining intervals, and appropriate type of parenthesis ''('', '')'', ''['' or '']'' depending on whether the interval is open or closed.) The function is increasing on The function is decreasing on

Explanation / Answer

f(x)=8x^5 - 7x^3 +1

f '(x)=40x^4 -21x^2

For maxima or minima, f '(x)=0

or,x^2(40x^2-21)=0

or, x=0, -Sqrt(21/40), Sqrt(21/40)

Now f "(x)=160x^3 -42x

f "(-Sqrt(21/40))<0 hence local Maxima at x=-Sqrt(21/40)

f "(Sqrt(21/40))>0,hence local Manima at x=Sqrt(21/40)

for function increasing f '(x)>0 hence interval (-infinity,-Sqrt(21/40) )U(Sqrt(21/40), infinity)

or function decreasing f '(x)<0 hence interval (-Sqrt(21/40),0 )U(0, Sqrt(21/40))

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