Suppose function f(x) and its derivative f'(x) is well-defined for all real-valued x. It is known that f(x) has only one stationary point x=1. Moreover, we know that f(x) indeed has a maximal point. Is it correct to conclude that x=1 is the maximal point for f(x)? Question 3 Select one: True False

At any x, if first order and second order derivatives of a function f is equal to 0 and greater than 0 respectively, then1 pointx is a local maximum of fx is a local minimum ofx is a stationary pointNone of the above

f(x) = xx has a stationary point at [1]

If y=f(x) and the function is known to attain a maximum point at x0, then at the point x0a.f’(x0) <0b.f’(x0) >0c.f’’(x0) >0d.f’’(x0) <0

Suppose that f : [a, b] → R is a continuous function that satisfies f (x) ∈ [a, b]for all x ∈ [a, b]. Prove that if f is differentiable on (a, b) and |f ′(x)| < 1 for all x ∈ (a, b),then f has exactly one fixed point. (Hint: We know that there is at least one fixed point byHomework 6. Try a proof by contradiction, and use the Mean Value Theorem.

Consider the function and fill in the spaces below.  By the First derivative test, the function has:a local maximum at the point Answer 1 Question 7, Answer 2 Question 7 and a local minimum at the point