Question

Use the Fundamental Theorem of Calculus to determine the one point of

inflection of the function
where a > 0. Give the exact value of the x-coordinate of the point of
inflection.

Answer 1

The Fundamental Theorem of Calculus can be used to determine the one point of inflection of a function. Follow these steps: find the antiderivative, evaluate it at two points, subtract the values, and solve for the x-coordinate.

Step-by-step explanation:

The Fundamental Theorem of Calculus can be used to determine the one point of inflection of a function. First, we need to find the antiderivative of the function. Then, we can evaluate the antiderivative at two points, and subtract the value at the smaller point from the value at the larger point to find the definite integral. Finally, we can solve the equation for the x-coordinate of the point of inflection.

For example, let's consider the function f(x) = x^3 - 3x^2 + 2. To find the one point of inflection, we can follow these steps:

1. Find the antiderivative of the function: F(x) = (1/4)x^4 - x^3 + 2x + C
2. Evaluate the antiderivative at two points: F(0) = 0 and F(1) = 1/4 - 1 + 2 + C, where C is the constant of integration.
3. Subtract the value at the smaller point from the value at the larger point: (1/4 - 1 + 2 + C) - 0 = 1/4 - 1 + 2 + C
4. Solve the equation to find the x-coordinate of the point of inflection: 1/4 - 1 + 2 + C = 0

By solving the equation, we can find the exact value of the x-coordinate of the point of inflection for the given function.