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Chapter 2: Problem 69

Find the derivatives from the left and from the right at \(x=1\) (if they exist). Is the function differentiable at \(x=1 ?\) \(f(x)=\left\\{\begin{array}{ll}(x-1)^{3}, & x \leq 1 \\ (x-1)^{2}, & x>1\end{array}\right.\)

Short Answer

Expert verified
The left derivative is 0, the right derivative is 0, and since they are equal, the function is differentiable at \(x=1\).

Step by step solution

01

Calculate the left derivative

Since \(f(x) = (x - 1)^3\) for \(x \leq 1\), the left derivative, denoted by \(f^{-}(1)\), can be computed as the derivative of \(f(x)\) at \(x=1\). The derivative of \(f(x) = (x - 1)^3\) is \(f'(x) = 3 \cdot (x - 1)^2\). Substituting \(x = 1\) gives \(f^{-}(1) = 3 \cdot (1 - 1)^2 = 0\).
02

Calculate the right derivative

Since \(f(x) = (x - 1)^2\) for \(x > 1\), the right derivative, denoted by \(f^{+}(1)\), can be calculated as the derivative of \(f(x)\) at \(x=1\). The derivative of \(f(x) = (x - 1)^2\) is \(f'(x) = 2 \cdot (x - 1)\). Substituting \(x = 1\) gives \(f^{+}(1) = 2 \cdot (1 - 1) = 0\).
03

Compare the left and right derivatives

To determine if the function is differentiable at \(x=1\), compare the left and right derivatives. If they are equal, the function is differentiable. Here, since \(f^{-}(1) = 0\) and \(f^{+}(1) = 0\), they are equal, so the function is differentiable at \(x=1\).

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