Question

If f is a function such that
[(f(b)-f(a))/(b-a)]=2 , then which of the following statements must be true?

a.f(a) = f(b) = 2

b.The slope of the tangent line to the function at x = a is 2.

c.The average rate of change of the function on the interval [a, b] is 2

d.The linear approximation for f(x) at x = a is y = 2

Answer 1

so if you think about, the problem presents us with a function, f, and 2 different values of x that are put into f, which are a and b.
if f(b) and f(a) yield y values, and b and a are x values, doesn't that make the given formula the slope formula? see:
f(b) = y2, while f(a) = y1 --> b = x2, and a = x1, so the given formula becomes
y2 - y1 / x2 - x1 = 2
So some slope of f is equal to 2. This can narrow down your answer to either b or c. But if you choose b, you'll see that this doesn't account for x=b. Choosing c, you can understand that the slope is equal to 2 where [a,b]; the average slope (or rate of change) in the interval [a,b] is 2