Determine Values of k and p for Continuity and Differentiability of a Piecewise Function at x = 1

Math Problem Statement

Solve C

For what values of k and p is the function f(x) both continuous and differentiable at x = 1?

Solution

To determine the values of $k$ and $p$ such that the function $f(x)$ is both continuous and differentiable at $x = 1$ , we need to ensure that:

Continuity: The left-hand limit $\lim_{x \to 1^-} f(x)$ and right-hand limit $\lim_{x \to 1^+} f(x)$ must be equal to $f(1)$ .
Differentiability: The derivatives from the left and right at $x = 1$ must also be equal.

The function $f(x)$ is defined as:

2 - x^2, & x \leq 1 \\ kx + p, & x > 1 \end{cases}$$ ### Step 1: Continuity at $$x = 1$$ To ensure continuity, we need: $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$$ #### Left-hand limit as $$x \to 1^-$$: $$\lim_{x \to 1^-} f(x) = 2 - (1)^2 = 2 - 1 = 1$$ #### Right-hand limit as $$x \to 1^+$$: For $$x > 1$$, $$f(x) = kx + p$$. So: $$\lim_{x \to 1^+} f(x) = k(1) + p = k + p$$ To satisfy continuity, these limits must be equal: $$1 = k + p$$ ### Step 2: Differentiability at $$x = 1$$ To ensure differentiability, we also need the derivatives from the left and right at $$x = 1$$ to be equal. #### Derivative from the left (using $$f(x) = 2 - x^2$$ for $$x \leq 1$$): $$f'(x) = -2x$$ Thus, $$\lim_{x \to 1^-} f'(x) = -2(1) = -2$$ #### Derivative from the right (using $$f(x) = kx + p$$ for $$x > 1$$): $$f'(x) = k$$ Thus, $$\lim_{x \to 1^+} f'(x) = k$$ For differentiability, we need: $$-2 = k$$ ### Solving for $$k$$ and $$p$$ From continuity, we found: $$1 = k + p$$ Substituting $$k = -2$$ into this equation: $$1 = -2 + p$$ $$p = 3$$ ### Conclusion The values of $$k$$ and $$p$$ that make $$f(x)$$ both continuous and differentiable at $$x = 1$$ are: $$k = -2 \quad \text{and} \quad p = 3$$ Would you like further details or have any questions? --- Here are five related questions to consider: 1. What if the function were continuous but not differentiable at $$x = 1$$? How would that affect $$k$$ and $$p$$? 2. Can you verify the differentiability of $$f(x)$$ at other points, such as $$x = 0$$? 3. What happens if $$k$$ or $$p$$ take values different from those we calculated? 4. How would the analysis change if the function were defined differently for $$x < 1$$? 5. Can we generalize this approach for other piecewise functions with quadratic and linear parts? **Tip:** For any piecewise function, always start by checking continuity before differentiability at the point where the definition changes.

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Math Problem Analysis

Mathematical Concepts

Limits
Continuity
Differentiability
Piecewise Functions

Formulas

Continuity condition: lim(x -> c^-) f(x) = lim(x -> c^+) f(x) = f(c)
Differentiability condition: lim(x -> c^-) f'(x) = lim(x -> c^+) f'(x)

Theorems

Limit Definition of Continuity
Differentiability Implies Continuity

Suitable Grade Level

High School (Calculus)

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