Word Count : 662

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Let's solve this step-by-step: ### Step 1: Understand the function definition The function \( f(x) \) is piecewise defined as: \[ f(x) = \begin{cases} -x^2 & \text{for } x \leq 0 \\ 4 - 5x & \text{for } 0 < x \leq 1 \\ 3x - 4x^2 & \text{for } 1 < x \leq 2 \\ -12x + 2x & \text{for } x > 2 \end{cases} \] We need to determine the points of discontinuity and their nature, and check if the function is differentiable at \( x = 1 \) (since we are asked to check for differentiability at \( x = 0.1 \), we will first investigate \( x = 1 \), as it’s a potential discontinuity). ### Step 2: Identify the potential points of discontinuity Discontinuities can occur at _______ ____ _______ ___ __________ ____ ________ _____ ____.
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