Question
a) Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:
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Also check whether the function f is derivable at x = 1.
Answer :
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Skip in 10
We are asked to analyze the discontinuities and differentiability of the piecewise function: [ f(x) = \begin{cases} -x^2, & x \le 0 \ 4-5x, & 0 < x \le 1 \ 3x-4x^2, & 1 < x \le 2 \ -12x + 2x, & x > 2 \end{cases} ] --- Step 1: Check points of discontinuity Discontinuities can occur at the boundaries of the piecewise definition: (x = 0, 1, 2). 1. At (x = 0): * Left-hand limit: (\lim_{x\to 0^-} f(x) = \lim_{x\to 0^-} (-x^2) = 0) * Right-hand limit: (\lim_{x\to 0^+} f(x) = \lim_{x\to 0^+} (4-5x) = 4) * Since (\lim_{x\to 0^-} f(x) \neq \lim_{x\to 0^+} f(x)), there is a jump discontinuity at (x = 0). 2. At (x = 1): * Left-hand limit: (\lim_{x\to 1^-} f(x) = \lim_{x\to
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