Math Problem Statement
Determine the values of a and b which would result in the function f(x) being differentiable at X = 1. f() = Зах + 1 3622
- 3x - 9 for for X < 1 X > 1
Solution
To ensure the function is differentiable at , we must first ensure that it is continuous at and that the derivative from both sides at is equal.
Given:
\begin{cases} 3ax + 1 & \text{for } x < 1, \\ 3bx^2 - 3x - 9 & \text{for } x > 1. \end{cases}$$ ### Step 1: Continuity at $$x = 1$$ For $$f(x)$$ to be continuous at $$x = 1$$: $$\lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^+}} f(x) = f(1)$$ **Left-hand limit:** $$\lim_{{x \to 1^-}} f(x) = 3a(1) + 1 = 3a + 1$$ **Right-hand limit:** $$\lim_{{x \to 1^+}} f(x) = 3b(1)^2 - 3(1) - 9 = 3b - 3 - 9 = 3b - 12$$ Setting these equal for continuity: $$3a + 1 = 3b - 12$$ $$3a - 3b = -13 \quad \text{(Equation 1)}$$ ### Step 2: Differentiability at $$x = 1$$ For $$f(x)$$ to be differentiable at $$x = 1$$, the derivatives from both sides must also be equal: **Left-hand derivative:** $$f'(x) \text{ for } x < 1: \quad \frac{d}{dx} (3ax + 1) = 3a$$ **Right-hand derivative:** $$f'(x) \text{ for } x > 1: \quad \frac{d}{dx} (3bx^2 - 3x - 9) = 6bx - 3$$ At $$x = 1$$: $$f'(1) = 6b(1) - 3 = 6b - 3$$ Setting these equal for differentiability: $$3a = 6b - 3$$ $$3a - 6b = -3 \quad \text{(Equation 2)}$$ ### Step 3: Solving the System of Equations We now solve the system of equations: 1. $$3a - 3b = -13$$ 2. $$3a - 6b = -3$$ Subtract Equation 1 from Equation 2: $$(3a - 6b) - (3a - 3b) = -3 - (-13)$$ $$-3b = 10$$ $$b = -\frac{10}{3}$$ Substitute $$b = -\frac{10}{3}$$ into Equation 1: $$3a - 3\left(-\frac{10}{3}\right) = -13$$ $$3a + 10 = -13$$ $$3a = -23$$ $$a = -\frac{23}{3}$$ ### Final Answer The values of $$a$$ and $$b$$ that make the function $$f(x)$$ differentiable at $$x = 1$$ are: $$a = -\frac{23}{3}, \quad b = -\frac{10}{3}$$ --- Do you have any questions or would you like more details on any step? ### Related Questions: 1. How do you check if a function is continuous at a point? 2. What is the geometric interpretation of differentiability at a point? 3. How can the concept of limits help in understanding the continuity of functions? 4. What is the significance of having equal derivatives from both sides at a point? 5. Can you give an example of a function that is continuous but not differentiable at a point? ### Tip: When solving systems of equations, it's helpful to simplify one equation at a time to make substitution or elimination straightforward.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Differentiability
Limits
System of Equations
Formulas
Derivative formulas
Theorems
Differentiability implies continuity
Suitable Grade Level
Undergraduate
Related Recommendation
Continuity and Differentiability of a Piecewise Function at a Point
Find Values of a and b for Continuity and Differentiability in a Piecewise Function
Finding Constants for Continuity and Differentiability in a Piecewise Function
Find a and b for Continuity and Differentiability in a Piecewise Function
Ensuring Continuity and Differentiability of a Piecewise Function