Find the value of c such that the function f(x) = { x^3 - 3x + 2, x < 1; c, x

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Question: Find the value of c such that the function f(x) = { x^3 - 3x + 2, x < 1; c, x = 1; x^2 + 1, x > 1 is continuous at x = 1.

Options:

Correct Answer: 2

Solution:

To ensure continuity at x = 1, we set the left limit (1 - 3 + 2 = 0) equal to the right limit (1 + 1 = 2), leading to c = 2.

Find the value of c such that the function f(x) = { x^3 - 3x + 2, x < 1; c, x = 1; x^2 + 1, x > 1 is continuous at x = 1.

Find the value of c such that the function f(x) = { x^3 - 3x + 2, x < 1; c, x = 1; x^2 + 1, x > 1 is continuous at x = 1.

Step 1: Understand that for a function to be continuous at a point, the left limit, right limit, and the function value at that point must all be equal.
Step 2: Identify the function f(x) and the point of interest, which is x = 1.
Step 3: Calculate the left limit as x approaches 1 from the left (x < 1). This means using the part of the function f(x) = x^3 - 3x + 2.
Step 4: Substitute x = 1 into the left part: f(1) = 1^3 - 3(1) + 2 = 1 - 3 + 2 = 0.
Step 5: Calculate the right limit as x approaches 1 from the right (x > 1). This means using the part of the function f(x) = x^2 + 1.
Step 6: Substitute x = 1 into the right part: f(1) = 1^2 + 1 = 1 + 1 = 2.
Step 7: Set the left limit equal to the right limit to find c: 0 (left limit) must equal c (value at x = 1) and also equal 2 (right limit).
Step 8: Since the left limit is 0 and the right limit is 2, we need to set c = 2 to make the function continuous at x = 1.

Continuity of Functions – Understanding the conditions for a function to be continuous at a point, specifically ensuring that the left-hand limit, right-hand limit, and the function value at that point are equal.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value, and determining how to handle transitions between these expressions.