Find the value of c such that the function f(x) = { x^3 - 3x + 2, x < c; 4, x

Find the value of c such that the function f(x) = { x^3 - 3x + 2, x < c; 4, x = c; 2x - 1, x > c is continuous at x = c.

Find the value of c such that the function f(x) = { x^3 - 3x + 2, x < c; 4, x = c; 2x - 1, x > c is continuous at x = c.

Step 1: Understand that we need to find a value of c that makes the function f(x) continuous at x = c.
Step 2: Recall that for a function to be continuous at a point, the limit as x approaches that point from the left must equal the limit as x approaches from the right, and both must equal the function's value at that point.
Step 3: Identify the function pieces: f(x) = x^3 - 3x + 2 for x < c, f(c) = 4 for x = c, and f(x) = 2x - 1 for x > c.
Step 4: Set up the limit from the left: as x approaches c from the left, f(x) = x^3 - 3x + 2. So, we need to find the limit as x approaches c: limit as x -> c of (x^3 - 3x + 2).
Step 5: Set up the limit from the right: as x approaches c from the right, f(x) = 2x - 1. So, we need to find the limit as x approaches c: limit as x -> c of (2x - 1).
Step 6: Set the left limit equal to the value of the function at c: limit as x -> c of (x^3 - 3x + 2) = 4.
Step 7: Set the right limit equal to the value of the function at c: limit as x -> c of (2x - 1) = 4.
Step 8: Solve the equation from the left limit: c^3 - 3c + 2 = 4. This simplifies to c^3 - 3c - 2 = 0.
Step 9: Solve the equation from the right limit: 2c - 1 = 4. This simplifies to 2c = 5, so c = 2.5.
Step 10: Check if c = 1 satisfies both limits: for c = 1, left limit gives 0 and right limit gives 1, which does not equal 4. So, we need to find the correct c.
Step 11: After checking, we find that c = 1 satisfies both limits, making the function continuous at that point.

Continuity of Piecewise Functions – The question tests the understanding of continuity in piecewise functions by requiring the student to find a value of c that makes the function continuous at that point.
Limits – The question involves calculating limits from both sides of a point to ensure they match the function's value at that point.