Question

In your lab, a substance's temperature has been observed to follow the function f(x) = (x − 1)3 + 9. The point at which the function changes curvature from concave down to concave up is where the substance changes from a solid to a liquid. What is the point where this function changes curvature from concave down to concave up?

Hint: The point is labeled in the picture.

Answer 1

The point of inflection is (1,9)

Explanation:

We have following given function

`f(x)=(x-1)^(3) +9`

The point at which the function changes its curvature is defined by the point of inflection.

To find point of inflection we set 2nd derivative to 0

`f''(x)= 0`

The first derivative is given by

`f'(x)= (d)/(dx) [(x-1)^(3) +9]`

`f'(x)= 3(x-1)^(2)` ( using chain rule and derivative of constant is 0)

now again we take 2nd derivative

`f''(x)=3(2(x-1))`

`f''(x)=6(x-1)`

now we equate 2nd derivative to 0

`6(x-1)=0\\x-1=0\\x=1`

hence point of inflection is at x=1

now we find y coordinate of point of inflection by plugging x=1 in f(x)

`y=f(1)=(1-1)^(3) +9 =9`

Hence the point of inflection is (1,9)

Answer 2

The point of change is the point (1,9)

Explanation:

The first derivative of the function equated to zero will give you the point where the function changes.

Therefore, to solve this problem find the first derivative of f (x)

`f '(x) = 3*3(x-1)^(3-1)\\\\f '(x) = 9(x-1)^2`

Now we equate the derivative to 0.

`9 (x-1)^ 2 = 0\\\\x = 1`

Then the derivative of the function is equal to 0 when x = 1. This means that the concavity of the function changes in x = 1.

When
`x = 1, y = 3(1-1) ^ 3 +9\\\\x =1, y = 9.`

Then the point of change is the point (1,9)