Question

Find the points of inflection of the qraph of the function.

f(x)=(x-6)³(x-1)
smaller x value (x,y) = __

Answer 1

To find the points of inflection of the function f(x) = (x - 6)³(x - 1), find the second derivative and set it equal to zero. Solve for x to find the x-values of the inflection points.

Step-by-step explanation:

To find the points of inflection of the graph of the function f(x) = (x - 6)³(x - 1), we need to determine the x-values where the concavity changes. A point of inflection occurs when the second derivative of the function changes sign. Let's start by finding the second derivative of f(x).

First, find the first derivative: f'(x) = 3(x - 6)²(x - 1) + (x - 6)³.

Next, find the second derivative: f''(x) = 6(x - 6)(x - 1) + 3(x - 6)².

Set f''(x) = 0 and solve for x to find the x-values of the inflection points.

6(x - 6)(x - 1) + 3(x - 6)² = 0.

Simplify and solve for x to find the x-values of the inflection points.