let (f) be the function defined above. for what values of (b) is (f) continuous at (x = 2)?\na 0.508 only\nb…

Explanation:

Step1: Recall the continuity condition

For a function to be continuous at (x = a), (\lim_{x\rightarrow a^{-}}f(x)=\lim_{x\rightarrow a^{+}}f(x)=f(a)). Here (a = 2), so (\lim_{x\rightarrow 2^{-}}e^{bx}=\lim_{x\rightarrow 2^{+}}(1.5x + b)).

Step2: Calculate the left - hand limit

(\lim_{x\rightarrow 2^{-}}e^{bx}=e^{2b}).

Step3: Calculate the right - hand limit

(\lim_{x\rightarrow 2^{+}}(1.5x + b)=1.5\times2 + b=3 + b).

Step4: Set the left - hand and right - hand limits equal

Set (e^{2b}=3 + b). Let (y = e^{2b}-b - 3). We can use a numerical method (such as Newton - Raphson method). The derivative of (y) with respect to (b) is (y^\prime=2e^{2b}-1). Starting with an initial guess, we find the root of the equation (e^{2b}-b - 3 = 0). By using a graphing utility or a calculator with equation - solving capabilities, we find that (b\approx0.508).

Answer:

A. 0.508 only