# What are the points of inflection, if any, of #f(x)=2x(2x-3)^3 #?

To determine the nature of these points we need to examine the second derivative to see if f'(x) is increasing (min), decreasing (max) or 0 (inflection):

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To find the points of inflection of ( f(x) = 2x(2x-3)^3 ), you first need to find the second derivative of the function. Then, set the second derivative equal to zero and solve for ( x ). Finally, determine whether these solutions correspond to points of inflection by analyzing the concavity of the function.

First derivative: [ f'(x) = 2(2x-3)^3 + 2x \cdot 3(2x-3)^2 ]

Second derivative: [ f''(x) = 6(2x-3)^2 + 2(2x-3)^2 + 12x(2x-3) ]

Setting the second derivative equal to zero: [ 6(2x-3)^2 + 2(2x-3)^2 + 12x(2x-3) = 0 ]

Solving this equation will give you the potential points of inflection. After finding these ( x ) values, you can determine whether they correspond to points of inflection by checking the concavity of the function around these points using the second derivative test.

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