# How do I find concavity and points of inflection for #y = 3x^5 - 5x^3#?

##### 1 Answer

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#### Explanation:

Given -

#y=3x^5-5x^3#

Find the first derivative -

#dy/dx=15x^4-15x^2#

#dy/dx =0 =>15x^4-15x^2=0 #

Then -

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

#15x^2=0#

#x=0#

#x^2-1=0#

#x^2=1#

#x=+-1#

#x=1#

#x=-1#

Find the second derivative -

#(d^2x)/(dx^2)=60x^3-30x#

At

#(d^2y)/(dx^2)=60(0)^3-30(0)=0#

The value of the function -

#y=3(0)^5-5(0)^3=0#

At

#dy/dx=0; (d^2y)/(dx^2)=0#

Hence there is a point of inflection at

At

#(d^2y)/(dx^2)=60(1)^3-30(1)=60-30=30>0#

The value of the function -

#y=3(1)^5-5(1)^3=03-5=-2#

At

#dy/dx=0; (d^2y)/(dx^2)>0#

Hence there is a minimum at

The curve is concave upwards

At

#(d^2y)/(dx^2)=60(-1)^3-30(-1)=-60+30=-30<0#

The value of the function -

#y=3(-1)^5-5(-1)^3=-3+5=2#

At

#dy/dx=0; (d^2y)/(dx^2)<0#

Hence there is a Maximum at

The curve is concave downwards.