Math Problem Statement
Let f(x) be a piecewise function defined as f(x) = cx + d for x ≤ 2, and f(x) = x^2 - cx for x > 2. If the function is differentiable at x = 2, what is the value of c + d?
Solution
We are given a piecewise function defined as follows:
\begin{cases} cx + d & \text{for } x \leq 2 \\ x^2 - cx & \text{for } x > 2 \end{cases}$$ where $$c$$ and $$d$$ are constants. We are also told that $$f$$ is differentiable at $$x = 2$$, and we need to find the value of $$c + d$$. ### Step 1: Continuity at $$x = 2$$ For the function to be differentiable at $$x = 2$$, it must first be continuous at $$x = 2$$. This means the two expressions for $$f(x)$$ must give the same value at $$x = 2$$. Thus, we set the two expressions for $$f(x)$$ equal at $$x = 2$$: $$f(2) = c(2) + d = 2c + d$$ $$f(2) = 2^2 - c(2) = 4 - 2c$$ Equating these two values gives the equation: $$2c + d = 4 - 2c$$ Solving for $$d$$: $$2c + 2c + d = 4 \quad \Rightarrow \quad 4c + d = 4$$ $$d = 4 - 4c$$ ### Step 2: Differentiability at $$x = 2$$ Next, for the function to be differentiable at $$x = 2$$, the derivatives of the two expressions must also be equal at $$x = 2$$. The derivative of $$f(x)$$ for $$x \leq 2$$ is: $$f'(x) = c$$ The derivative of $$f(x)$$ for $$x > 2$$ is: $$f'(x) = 2x - c$$ At $$x = 2$$, this becomes: $$f'(2) = 2(2) - c = 4 - c$$ Equating the two derivatives at $$x = 2$$: $$c = 4 - c$$ Solving for $$c$$: $$2c = 4 \quad \Rightarrow \quad c = 2$$ ### Step 3: Finding $$d$$ Substitute $$c = 2$$ into the equation $$d = 4 - 4c$$: $$d = 4 - 4(2) = 4 - 8 = -4$$ ### Step 4: Finding $$c + d$$ Now that we have $$c = 2$$ and $$d = -4$$, we can find $$c + d$$: $$c + d = 2 + (-4) = -2$$ Thus, the value of $$c + d$$ is $$\boxed{-2}$$, which corresponds to option **B**. Would you like further details or explanations? Here are five related questions: 1. What is the condition for a function to be differentiable at a point? 2. How would the solution change if $$f(x)$$ was not differentiable at $$x = 2$$? 3. Can you explain how to find limits of piecewise functions? 4. What is the significance of the first derivative at a point? 5. How does continuity relate to differentiability? **Tip**: Always check both the continuity and differentiability conditions when dealing with piecewise functions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Differentiability
Continuity
Piecewise Functions
Formulas
f'(x) for cx + d is c
f'(x) for x^2 - cx is 2x - c
Theorems
Differentiability at a point requires both continuity and equal derivatives from both sides.
Suitable Grade Level
Grades 11-12
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