Question

If the function $$f(x) = \begin{array}{cc} 3ax + b, & \text{for } x < 1 \\ 11, & \text{for } x = 1 \\ 5ax - 2b, & \text{for } x > 1 \end{array}$$ is continuous at $x = 1$, then the values of $a$ and $b$ are

• A. $a = 2, b = 3$
• B. $a = 3, b = 3$
• C. $a = 2, b = 2$
• D. $a = 3, b = 2$

Hint:

To solve multiple-choice limit systems even faster, plug the options directly into the linear boundary constraints! Testing option (D) gives $3(3) + 2 = 11$ and $5(3) - 2(2) = 15 - 4 = 11$. Since both conditions match the central value of $11$ perfectly, it confirms the correct option instantly without any manual elimination steps.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a piecewise defined function containing two unknown parameters $a$ and $b$. The function is specified to be continuous at the boundary point $x = 1$, and we need to solve for the values of $a$ and $b$.

Step 2: Key Formula or Approach:
For a function to be continuous at a point $x = c$, the Left-Hand Limit (LHL), Right-Hand Limit (RHL), and the value of the function at that point must all be equal: $$\lim_{x \rightarrow c^-} f(x) = \lim_{x \rightarrow c^+} f(x) = f(c)$$

Step 3: Detailed Explanation:
Let's find the values of the individual limit components at $x = 1$:

• Left-Hand Limit (LHL): Use the expression for $x < 1$: $$\text{LHL} = \lim_{x \rightarrow 1^-} (3ax + b) = 3a(1) + b = 3a + b$$

• Right-Hand Limit (RHL): Use the expression for $x > 1$: $$\text{RHL} = \lim_{x \rightarrow 1^+} (5ax - 2b) = 5a(1) - 2b = 5a - 2b$$

• Value of the function: Given explicitly as $f(1) = 11$.
Since the function is continuous at $x = 1$, equate the LHL and RHL to the function value to form a system of simultaneous linear equations: align 3a + b &= 11
5a - 2b &= 11 align Let's solve this system. Multiply equation (1) by 2 to prepare for elimination: $$6a + 2b = 22$$ Add this new equation directly to equation (2): $$(6a + 2b) + (5a - 2b) = 22 + 11$$ $$11a = 33 \implies a = 3$$ Substitute $a = 3$ back into equation (1) to find $b$: $$3(3) + b = 11 \implies 9 + b = 11 \implies b = 2$$ The solution values are $a = 3$ and $b = 2$.

Step 4: Final Answer:
The values of the parameters are $a = 3$ and $b = 2$, which corresponds to option (D).