Math Problem Statement
Solve problem 1
The function f(x) = { x^2 - ax if x <= 1, ax + b if x > 1, where a and b are constants. If f is differentiable at x = 1, then a + b = ?
Solution
To solve problem 1, we need to determine the constants and so that the function is differentiable at . The function is defined as:
\begin{cases} x^2 - ax & \text{if } x \leq 1 \\ ax + b & \text{if } x > 1 \end{cases}$$ For $$f(x)$$ to be differentiable at $$x = 1$$, it must be continuous and have the same derivative from both sides at $$x = 1$$. ### Step 1: Continuity at $$x = 1$$ The continuity condition requires: $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1).$$ Calculating these limits: 1. From the left ($$x \leq 1$$): $$f(1) = 1^2 - a \cdot 1 = 1 - a.$$ 2. From the right ($$x > 1$$): $$f(1) = a \cdot 1 + b = a + b.$$ Equating both sides for continuity: $$1 - a = a + b. \quad \text{(Equation 1)}$$ ### Step 2: Differentiability at $$x = 1$$ The differentiability condition requires: $$\lim_{x \to 1^-} f'(x) = \lim_{x \to 1^+} f'(x).$$ Finding the derivatives: 1. For $$x \leq 1$$: $$f'(x) = \frac{d}{dx}(x^2 - ax) = 2x - a.$$ So, $$f'(1) = 2 \cdot 1 - a = 2 - a.$$ 2. For $$x > 1$$: $$f'(x) = \frac{d}{dx}(ax + b) = a.$$ Equating both derivatives for differentiability: $$2 - a = a. \quad \text{(Equation 2)}$$ ### Step 3: Solving the Equations Now, solve Equations 1 and 2 simultaneously: From Equation 2: $$2 - a = a \implies 2 = 2a \implies a = 1.$$ Substituting $$a = 1$$ into Equation 1: $$1 - 1 = 1 + b \implies 0 = 1 + b \implies b = -1.$$ Therefore, $$a + b = 1 + (-1) = 0$$. ### Final Answer: $$\boxed{0 \text{ (Option C)}}$$ Would you like to see more details or have any questions? ### Related Questions: 1. What is the difference between continuity and differentiability? 2. How do you check if a function is differentiable at a point? 3. Can a function be continuous but not differentiable? Provide an example. 4. What happens if $$a$$ and $$b$$ do not satisfy the differentiability condition? 5. How can you graphically verify if a function is differentiable at a given point? ### Tip: Always ensure continuity before checking differentiability; differentiability implies continuity, but not the other way around!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Differentiability
Piecewise Functions
Limits
Formulas
f(x) = x^2 - ax
f(x) = ax + b
f'(x) = 2x - a
f'(x) = a
Theorems
Differentiability implies continuity
Limit definition of derivative
Suitable Grade Level
Grade 12, AP Calculus
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