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

Answer:

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

Step-by-step 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)

Answer 2

Answer:

The point of inflection is (1,9)

Step-by-step 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

$f'(x)= \frac{d}{dx} [(x-1)^{3} +9]$

$f'(x)= 3(x-1)^{2}$     ( 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)