Consider the function y = x4 + 1.On the interval (−∞, 0) the derivative y′ is(A) negative and the function is decreasing(B) positive and the function is increasing(enter A/B)On the interval (0, ∞) the derivative y′ is(A) negative and the function is decreasing(B) positive and the function is increasing(enter A/B)

How do you determine where f is increasing or decreasing?If f  ′(x) 0 on an interval, then f is increasing on that interval.If f  ′(x) 0 on an interval, then f is decreasing on that interval.

Suppose f '' is continuous on (−∞, ∞).(a) If f '(3) = 0 and f ''(3) = −5, what can you say about f ?At x = 3, f has a local maximum.At x = 3, f has a local minimum.    At x = 3, f has neither a maximum nor a minimum.More information is needed to determine if f has a maximum or minimum at x = 3.(b) If f '(4) = 0 and f ''(4) = 0, what can you say about f ?At x = 4, f has a local maximum.At x = 4, f has a local minimum.    At x = 4, f has neither a maximum nor a minimum.More information is needed to determine if f has a maximum or minimum at x = 4.

Given f(1) = 4 and f '(1) = −5, what is the slope of the graph of f at the point (1, 4)?

Use the graph to determine the open intervals over which  is increasing, decreasing, or constant, then determine all the local minimum and maximum values on the graph.a.)Decreasing on Increasing on Local minimum is -2 at  and Local maximum is -1 at b.)Decreasing on Increasing on Local minimum is 0 at and 2 at Local maximum is 1 at c.)Decreasing on Increasing on Local minimum is 0 at and 2 at Local maximum is 1 at d.)Decreasing on Increasing on Local minimum is -2 at and Local maximum is -1 at

Suppose that f : [0, 4] → R is continuous on [0, 4], is differentiable on (0, 4) and thatf (0) = 0, f (1) = 3, f (4) = −3.Show that there exists c ∈ (0, 4) such that f ′(c) = −1