Suppose f and g are continuous functions such that g(2) = 4 and lim x → 2 [3f(x) + f(x)g(x)] = 28. Find f(2).

To find f(2), we need to use the given information and the properties of limits.

We know that the limit of a sum is the sum of the limits, and the limit of a product is the product of the limits. So we can rewrite the limit as follows:

lim x → 2 [3f(x) + f(x)g(x)] = lim x → 2 3f(x) + lim x → 2 f(x)g(x)

We also know that the limit of a constant times a function is the constant times the limit of the function. So we can rewrite the limit again as follows:

lim x → 2 3f(x) + lim x → 2 f(x)g(x) = 3 * lim x → 2 f(x) + lim x → 2 f(x) * lim x → 2 g(x)

We are given that lim x → 2 [3f(x) + f(x)g(x)] = 28 and g(2) = 4. So we can substitute these values into the equation:

28 = 3 * lim x → 2 f(x) + lim x → 2 f(x) * 4

Simplify the equation:

28 = 3f(2) + 4f(2)

Combine like terms:

28 = 7f(2)

Finally, solve for f(2):

f(2) = 28 / 7 = 4

So, f(2) = 4.