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^2 + 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^2 + 1, x >= 1 is continuous at x = 1.
Step 1: Identify the function f(x) which has two parts: f(x) = x^3 - 3x + p for x < 1 and f(x) = 2x^2 + 1 for x >= 1.
Step 2: To find the value of p that makes the function continuous at x = 1, we need to ensure that the two parts of the function equal each other at x = 1.
Step 3: Calculate the value of the first part of the function at x = 1: f(1) = 1^3 - 3(1) + p = 1 - 3 + p = p - 2.
Step 4: Calculate the value of the second part of the function at x = 1: f(1) = 2(1)^2 + 1 = 2 + 1 = 3.
Step 5: Set the two results equal to each other to ensure continuity: p - 2 = 3.
Step 6: Solve for p: p - 2 + 2 = 3 + 2, which gives p = 5.
Continuity of Piecewise Functions – The question tests the understanding of continuity at a point for piecewise functions, requiring the evaluation of limits from both sides and ensuring they match the function value at that point.
Limit Evaluation – It involves calculating the limit of the function as x approaches 1 from both sides and setting them equal to ensure continuity.
