Determine the value of p for which the function f(x) = { x^3 - 3x + p, x < 1;
Practice Questions
Q1
Determine the value of p for which the function f(x) = { x^3 - 3x + p, x < 1; 2x + 1, x >= 1 is continuous at x = 1.
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Questions & Step-by-Step Solutions
Determine the value of p for which the function f(x) = { x^3 - 3x + p, x < 1; 2x + 1, x >= 1 is continuous at x = 1.
Step 1: Understand that we need to find the value of p so that the function f(x) is continuous at x = 1.
Step 2: Recall that for a function to be continuous at a point, the left-hand limit and the right-hand limit at that point must be equal to the function's value at that point.
Step 3: Identify the two parts of the function: f(x) = x^3 - 3x + p for x < 1 and f(x) = 2x + 1 for x >= 1.
Step 4: Calculate the left-hand limit as x approaches 1 from the left (x < 1): f(1) = 1^3 - 3(1) + p = 1 - 3 + p = p - 2.
Step 5: Calculate the right-hand limit as x approaches 1 from the right (x >= 1): f(1) = 2(1) + 1 = 2 + 1 = 3.
Step 6: Set the left-hand limit equal to the right-hand limit: p - 2 = 3.
Step 7: Solve for p: p - 2 = 3 means p = 3 + 2, which gives p = 5.
Step 8: Verify that the function is continuous by checking if both limits are equal at x = 1.
Continuity of Piecewise Functions – The question tests the understanding of how to ensure continuity at a point for piecewise functions by equating the limits from both sides.
Limit Evaluation – It requires evaluating the limits of the function as x approaches the point of interest (x = 1) from both sides.
Solving for Parameters – The question involves solving for a parameter (p) that ensures the function is continuous.
