Final answer:
The question asks for the value of 'b' in a function with a typo. To find 'b', derivatives must be taken and evaluated at the given points, but the typo prevents an exact solution. Additional information or clarification on the function's format is required.
Step-by-step explanation:
The student's question pertains to finding the value of the constant 'b' in the function f(x) = 3x + ar' +bx+k, given that the function has a local minimum at x = -2 and a point of inflection at x = -3. To find the value of 'b', we'll need to use the information about the local minimum and point of inflection. A local minimum implies that the first derivative 'f'(x) will be zero at x = -2, and the second derivative 'f''(x) will be positive. A point of inflection implies that the second derivative 'f''(x) will be zero at x = -3.
First, we would differentiate the function to get 'f'(x) and then 'f''(x). However, as the function involves a term with ar', there might be a typo as it is not standard notation. Assuming the term should be 'ar^2', we can proceed with differentiation: 'f'(x) = 3 + 2ar + b, and 'f''(x) = 2a. By using the values 'f'(x) = 0 at x = -2 and 'f''(x) = 0 at x = -3, we can form a system of equations to solve for 'a' and 'b'. Unfortunately, with the information given and the typo in the equation, we cannot determine the exact value of 'b' without additional information or clarification of the function.