Math Problem Statement
Determine the values of a, b, and c such that the function f satisfies the hypotheses of the Mean Value Theorem on the interval [0, 3].
Solution
The problem asks to determine the values of , , and such that the function satisfies the hypotheses of the Mean Value Theorem (MVT) on the interval .
The function is defined piecewise:
\begin{cases} 3 & \text{if } x = 0 \\ ax + b & \text{if } 0 < x \leq 1 \\ x^2 + 2x + c & \text{if } 1 < x \leq 3 \end{cases}$$ To satisfy the Mean Value Theorem, the function must be: 1. Continuous on the closed interval $$[0, 3]$$. 2. Differentiable on the open interval $$(0, 3)$$. Let's go step-by-step to impose the conditions for continuity and differentiability. ### Step 1: Continuity at $$x = 1$$ For continuity at $$x = 1$$, the two pieces on either side of $$x = 1$$ must be equal at $$x = 1$$: $$ax + b = x^2 + 2x + c \quad \text{at} \quad x = 1$$ Substitute $$x = 1$$ into both expressions: $$a(1) + b = 1^2 + 2(1) + c$$ $$a + b = 1 + 2 + c$$ $$a + b = 3 + c \quad \text{(Equation 1)}$$ ### Step 2: Continuity at $$x = 0$$ For continuity at $$x = 0$$, the first piece must match the second piece at $$x = 0$$: $$f(0) = 3$$ Substitute $$x = 0$$ into the second piece: $$a(0) + b = 3$$ $$b = 3 \quad \text{(Equation 2)}$$ ### Step 3: Differentiability at $$x = 1$$ For differentiability at $$x = 1$$, the derivatives of the two pieces must match at $$x = 1$$. Derivative of $$ax + b$$ is: $$f'(x) = a$$ Derivative of $$x^2 + 2x + c$$ is: $$f'(x) = 2x + 2$$ At $$x = 1$$, the derivatives must be equal: $$a = 2(1) + 2$$ $$a = 4 \quad \text{(Equation 3)}$$ ### Step 4: Solve the System of Equations We now have three equations: 1. $$a + b = 3 + c$$ 2. $$b = 3$$ 3. $$a = 4$$ Substitute $$a = 4$$ and $$b = 3$$ into Equation 1: $$4 + 3 = 3 + c$$ $$7 = 3 + c$$ $$c = 4$$ ### Final Answer Thus, the values are: $$a = 4, \quad b = 3, \quad c = 4$$ Would you like further clarification or have any questions? ### Related Questions 1. What does the Mean Value Theorem state, and why is differentiability required? 2. Can a function be continuous but not differentiable? Provide an example. 3. How do you compute the derivative of a piecewise function? 4. What conditions need to be checked for the Mean Value Theorem besides differentiability? 5. How can you apply the Mean Value Theorem in practical scenarios? **Tip:** Continuity and differentiability are key properties when working with piecewise functions. Always check transitions at boundary points between different pieces.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
Differentiability
Mean Value Theorem
Formulas
ax + b for 0 < x ≤ 1
x² + 2x + c for 1 < x ≤ 3
Mean Value Theorem: f'(c) = (f(b) - f(a)) / (b - a)
Theorems
Mean Value Theorem
Suitable Grade Level
Grades 11-12
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