What are the points of inflection, if any, of #f(x)=x^4-5x^3+x^2

#?

Question

What are the points of inflection, if any, of #f(x)=x^4-5x^3+x^2

#?

Ezra Adams · Accepted Answer

#x=+-sqrt(5/2)#

The second derivative will be positive in the second case and negative in the first, meaning that the points of inflection are found at those points where the slope is changing from positive to negative (decreasing) or from negative to positive (increasing).

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

#f''(x)=12x^2-30#

This will change sign at the point where #x^2=30/12 = 5/2#

There are therefore two points of inflection, at #x=+-sqrt(5/2)#. Between these two values #f''(x)# is negative and elsewhere it is positive.

Cameron Alexander · Answer

undefined To find the points of inflection, we first need to find the second derivative of the function.

f''(x) = 12x^2 - 30x + 2

Next, we find the values of x where f''(x) = 0.

12x^2 - 30x + 2 = 0

Solving this quadratic equation gives us the values of x.

x = (15 ± √109)/6

Therefore, the points of inflection are (15 + √109)/6 and (15 - √109)/6.