$f(x)=\left\{\begin{array}{cc} x^{2}-4 x+4 & x<1 \\ 1 & x=1 \\ x^{2} & x>1 \end{array}\right.$ $\newline$ a) Study Differentiabilitu

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Evaluate definite integrals using the power rule

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f(x)=\left\{\begin{array}{cc} x^{2}-4 x+4 & x<1 \\ 1 & x=1 \\ x^{2} & x>1 \end{array}\right.

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a) Study Differentiabilitu

Check Continuity: We need to analyze the differentiability of the function $f(x)$ at the point $x = 1$ , since that is where the definition of the function changes. Differentiability at a point requires that the function is continuous at that point and that its derivative exists at that point. Let's first check for continuity at $x = 1$ .
Limit Calculation: For continuity at $x = 1$ , the limit of $f(x)$ as $x$ approaches $1$ from the left must equal the limit of $f(x)$ as $x$ approaches $1$ from the right, and both must equal $f(1)$ .
Continuity Verification: The limit of $f(x)$ as $x$ approaches $1$ from the left is the limit of the quadratic expression $x^2 - 4x + 4$ . This simplifies to $(1)^2 - 4(1) + 4 = 1 - 4 + 4 = 1$ .
Derivative Calculation: The limit of $f(x)$ as $x$ approaches $1$ from the right is the limit of the quadratic expression $x^2$ . This simplifies to $(1)^2 = 1$ .
Derivative Comparison: Since $f(1)$ is defined to be $1$ , we have that the left-hand limit, the right-hand limit, and the value of the function at $x = 1$ are all equal to $1$ . Therefore, the function is continuous at $x = 1$ .
Differentiability Conclusion: Next, we need to check if the derivative of $f(x)$ exists at $x = 1$ . We will find the derivative of the function on both sides of $x = 1$ and see if they are equal.
Differentiability Conclusion: Next, we need to check if the derivative of $f(x)$ exists at $x = 1$ . We will find the derivative of the function on both sides of $x = 1$ and see if they are equal.The derivative of $f(x)$ for $x < 1$ is the derivative of the quadratic expression $x^2 - 4x + 4$ , which is $2x - 4$ . Evaluating this at $x = 1$ gives us $2(1) - 4 = 2 - 4 = -2$ .
Differentiability Conclusion: Next, we need to check if the derivative of $f(x)$ exists at $x = 1$ . We will find the derivative of the function on both sides of $x = 1$ and see if they are equal.The derivative of $f(x)$ for $x < 1$ is the derivative of the quadratic expression $x^2 - 4x + 4$ , which is $2x - 4$ . Evaluating this at $x = 1$ gives us $2(1) - 4 = 2 - 4 = -2$ .The derivative of $f(x)$ for $x = 1$ $0$ is the derivative of the quadratic expression $x = 1$ $1$ , which is $x = 1$ $2$ . Evaluating this at $x = 1$ gives us $x = 1$ $4$ .
Differentiability Conclusion: Next, we need to check if the derivative of $f(x)$ exists at $x = 1$ . We will find the derivative of the function on both sides of $x = 1$ and see if they are equal.The derivative of $f(x)$ for $x < 1$ is the derivative of the quadratic expression $x^2 - 4x + 4$ , which is $2x - 4$ . Evaluating this at $x = 1$ gives us $2(1) - 4 = 2 - 4 = -2$ .The derivative of $f(x)$ for $x = 1$ $0$ is the derivative of the quadratic expression $x = 1$ $1$ , which is $x = 1$ $2$ . Evaluating this at $x = 1$ gives us $x = 1$ $4$ .Since the derivative from the left ( $x = 1$ $5$ ) is not equal to the derivative from the right ( $x = 1$ $6$ ), the function $f(x)$ is not differentiable at $x = 1$ .