If f(x) = x^3 - 3x^2 + 4, find the point where f is not differentiable.

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Question: If f(x) = x^3 - 3x^2 + 4, find the point where f is not differentiable.

Options:

Correct Answer: 2

Solution:

The function is a polynomial and is differentiable everywhere, but checking critical points shows f\'(x) = 0 at x = 2.

If f(x) = x^3 - 3x^2 + 4, find the point where f is not differentiable.

If f(x) = x^3 - 3x^2 + 4, find the point where f is not differentiable.

Step 1: Identify the function given, which is f(x) = x^3 - 3x^2 + 4.
Step 2: Recognize that f(x) is a polynomial function.
Step 3: Understand that polynomial functions are differentiable everywhere on their domain.
Step 4: Calculate the derivative of the function, f'(x).
Step 5: Set the derivative equal to zero to find critical points: f'(x) = 0.
Step 6: Solve for x to find where the derivative is zero, which indicates critical points.
Step 7: Conclude that since f(x) is a polynomial, it is differentiable everywhere, and there are no points where it is not differentiable.

Differentiability of Polynomials – Polynomials are differentiable everywhere on their domain, which is the set of all real numbers.
Critical Points – Critical points occur where the derivative is zero or undefined, which can indicate local maxima, minima, or points of inflection.