Determine the point at which the function f(x) = |x^2 - 4| is differentiable.

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Question: Determine the point at which the function f(x) = |x^2 - 4| is differentiable.

Options:

Correct Answer: x = -2

Solution:

f(x) is not differentiable at x = -2 and x = 2, but is differentiable everywhere else.

Determine the point at which the function f(x) = |x^2 - 4| is differentiable.

Determine the point at which the function f(x) = |x^2 - 4| is differentiable.

Step 1: Understand the function f(x) = |x^2 - 4|. This function involves an absolute value, which can create points where the function changes direction.
Step 2: Identify where the expression inside the absolute value equals zero. Set x^2 - 4 = 0 and solve for x.
Step 3: Solve the equation x^2 - 4 = 0. This gives x^2 = 4, so x = 2 and x = -2.
Step 4: These points (x = -2 and x = 2) are where the function f(x) could potentially not be differentiable because the absolute value can create sharp corners.
Step 5: Check the behavior of the function around x = -2 and x = 2. The function changes from one linear piece to another at these points.
Step 6: Conclude that f(x) is not differentiable at x = -2 and x = 2 because of the sharp corners, but it is differentiable everywhere else.

Absolute Value Functions – Understanding how absolute value affects differentiability, particularly at points where the expression inside the absolute value equals zero.
Differentiability – Recognizing that a function is not differentiable at points where it has sharp corners or cusps.
Critical Points – Identifying points where the derivative does not exist, which are often the roots of the expression inside the absolute value.