Home > Calculus > Determining Points of Inflection for a Function

What are the points of inflection, if any, of #f(x)= x^4-6x^3 #?

Answer 1

There exists a point of inflection at #(0,0)#

Points of inflections exist when #(d^2y)/dx^2=0# or when #f''(x)=0# We start by differentiating the function #f(x)=x^4-6x^3# we use #(d(x^n))/dx=nx^(n-1)# So #f'(x)=4x^3-18x^2# #f'(x)= 0# when #4x^3-18x^2=0# #2x^2(2x-9=0)# => #x=0# and #x=9/2# Then we calculate #f''(x)=12x^2-36x# we continue with the values of x obtained above #f''(0)=0# which is a point of inflection #f''(9/2)=12*(9/2)^2-36*(9/2)=12*9/4-18*9=27-162=-135# which is #<0# and is a minimum

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Answer 2

Emma Aldridge

To find the points of inflection of $f(x) = x^4 - 6x^3$ , we first find the second derivative and then set it equal to zero to solve for the values of $x$ .

First derivative:
$f'(x) = 4x^3 - 18x^2$

Second derivative:
$f''(x) = 12x^2 - 36x$

Setting the second derivative equal to zero:
$12x^2 - 36x = 0$

Factor out 12x:
$12x(x - 3) = 0$

Setting each factor equal to zero:
$12x = 0$ or $x - 3 = 0$

Solving for $x$ :
$x = 0$ or $x = 3$

Therefore, the points of inflection are $x = 0$ and $x = 3$ .

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.