solve the effect of the transformation on the graph

Question

1.   lets say that d ultiple Choice 1. Determine whether the given statements about a function f are true or false. Statement I: If lim ℎ→0 𝑓(𝑎+ℎ)−𝑓(𝑎) ℎ exists, then lim𝑥→𝑎 𝑓(𝑥) exists as well. Statement II: If f has an inflection point at 𝑥 = 𝑎, then 𝑓 ′′(𝑎) = 0. Statement III: If 𝑓 is continuous on [a,b], then ∫ 𝑓(𝑥)𝑑𝑥 = 𝑓(𝑏) − 𝑓(𝑎) 𝑏 𝑎 .

Answer 1

Statement I is true (if a function's derivative exists at x = a, then the function is continuous there).

Statement II is false. (A point of inflection is a where the graph of f changes concavity, so the 2nd derivative will change signs there (and often will = 0), but f(a) ≠ 0 necessarily.)

Statement III is false. If f continuous, then ∫_a^bf(x)dx = F(b) - F(a) , where F^'(x) = f(x).