Math Problem Statement
It should be differentiable Witherspoon x=2. So what is a and b to make this possible?
Solution
To ensure the function is differentiable at , we need to make sure the function is both continuous and has the same derivative from both sides of .
The piecewise function is given as:
\begin{cases} x^2 - 2x - 7 & \text{for } x < 2 \\ ax^2 + b & \text{for } x \geq 2 \end{cases}$$ ### 1. Continuity at $$x = 2$$ For the function to be continuous at $$x = 2$$, the left-hand limit as $$x$$ approaches 2 must equal the right-hand limit, and they both must equal $$f(2)$$. - For $$x < 2$$, $$f(x) = x^2 - 2x - 7$$. - Evaluate $$f(2)$$: $$f(2) = 2^2 - 2(2) - 7 = 4 - 4 - 7 = -7$$ - For $$x \geq 2$$, $$f(x) = ax^2 + b$$. - At $$x = 2$$, $$f(2) = a(2^2) + b = 4a + b$$. Set the two equal for continuity: $$4a + b = -7 \quad \text{(Equation 1)}$$ ### 2. Differentiability at $$x = 2$$ For differentiability, the left-hand derivative at $$x = 2$$ must equal the right-hand derivative at $$x = 2$$. - The derivative of $$f(x) = x^2 - 2x - 7$$ (for $$x < 2$$) is: $$f'(x) = 2x - 2$$ At $$x = 2$$, the left-hand derivative is: $$f'(2) = 2(2) - 2 = 4 - 2 = 2$$ - The derivative of $$f(x) = ax^2 + b$$ (for $$x \geq 2$$) is: $$f'(x) = 2ax$$ At $$x = 2$$, the right-hand derivative is: $$f'(2) = 2a(2) = 4a$$ Set the derivatives equal for differentiability: $$4a = 2 \quad \text{(Equation 2)}$$ ### 3. Solve the System of Equations From Equation 2: $$4a = 2 \quad \Rightarrow \quad a = \frac{2}{4} = \frac{1}{2}$$ Substitute $$a = \frac{1}{2}$$ into Equation 1: $$4\left(\frac{1}{2}\right) + b = -7 \quad \Rightarrow \quad 2 + b = -7 \quad \Rightarrow \quad b = -9$$ ### Conclusion: For the function to be differentiable at $$x = 2$$, the values of $$a$$ and $$b$$ must be: $$a = \frac{1}{2}, \quad b = -9$$ Would you like more details on how differentiability is ensured, or any questions on this process? ### Related Questions: 1. How do we check if a function is continuous at a point? 2. What are the conditions for a function to be differentiable at a point? 3. How do you calculate the left-hand and right-hand limits of a function? 4. Can a function be continuous but not differentiable at a point? If yes, provide examples. 5. How would the solution change if the function were defined differently for $$x > 2$$? ### Tip: When solving for differentiability, always ensure the function is continuous at the point in question first, as differentiability implies continuity but not the other way around!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Differentiability
Formulas
Continuity: Limit as x approaches 2 from the left must equal limit from the right
Differentiability: Derivatives from both sides at x = 2 must be equal
Theorems
For differentiability: A function must be continuous at the point in question and have equal derivatives from both sides
Suitable Grade Level
Grades 11-12 (Calculus level)
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