For which values of b is the function f defined as f(x) = e^{bx} for x < 2 and f(x) = 15x + b for x ≥ 2 continuous?

To determine the values of b that make the function f continuous at x = 2, we need to ensure that the left-hand limit and the right-hand limit of the function at this point are equal, as well as the value of the function at that point.

The function is defined in two pieces:

• f(x) = e^{bx} for x < 2
• f(x) = 15x + b for x ≥ 2

First, we will find the left-hand limit as x approaches 2:

lim_{x → 2^– f(x) = limx → 2^– e^bx = e^2b}

Next, we will find the right-hand limit as x approaches 2:

lim_{x → 2⁺ f(x) = limx → 2⁺ (15x + b) = 15(2) + b = 30 + b}

For the function to be continuous at x = 2, these two limits must be equal:

e^2b = 30 + b

Now, we have the equation:

e^2b – b – 30 = 0

To find the values of b, we can analyze the function g(b) = e^2b – b – 30. This function helps us find points where it crosses the x-axis (roots), indicating continuity. This transcendental equation typically requires numerical methods or graphical approaches to approximate solutions.

By carefully graphing or using numerical solvers, one can find appropriate values of b. You could also apply methods such as the Newton-Raphson method or bisection method to find more precise roots. In practice, common values often discovered through such analyses might yield specific intervals for b. Determining these solutions thoroughly involves testing values and ensuring that the condition for continuity holds true.

In summary, for the function f to be continuous at x = 2, we look for values of b that satisfy the equation e^2b = 30 + b. Numerical methods may be required to find the exact solutions.