Sketch the graph of a function that satisfies all of the given conditions
(a) $ f'(x) < 0 $ and $ f"(x) < 0 $ for all $ x $
(b) $ f'(x) > 0 $ and $ f"(x) > 0 $ for all $ x $
So we're told in part a that the derivative of our function F. Of X is less than zero. And also the second derivative of our function F. Of X is less than zero for all X. And what we want to do is use this information to try and sketch a potential graph of this function. And so what this tells us since our first derivative is less than zero for all X. That means that we're decreasing for the entirety of our function. So we're decreasing the entirety of our function. And since we know that the second derivative effects is less than zero, we also know that we are concave down for the entirety of our functions. The second derivative tells us about con cavity in the first derivative tells us about the slope or whether we're increasing or decreasing. And so what we want to do is we want to draw a graph that is both concave down and decreasing for all of its domain. So how we do this is I'm just going to start here and make sure and make sure that we're concave down and we're decreasing this entirety of this curve and then just put an air over there and an arrow like that where we're never actually going to get a horizontal um assume toad or slope or anything like that. We're just going to be concave down this entire way, but we're also decreasing. So this is what a sketch of the graph given these conditions would look like. And for part B we're told that F prime of X is greater than zero and F double prime of X is also greater than zero for all of the X. So in this case we're going to be concave up and increasing for the entirety of our function. So we want to draw a graph that is both concave up. Actually. Drawed on this type of graph that is concave up and increasing for the entirety of the function. So I'm just going to kind of start like like this and go like that and make sure that we're concave up so were curved upwards like a U. Shaped A upwards you shape. And we're also increasing for the entirety of our function. So a graph like this could be a potential sketch for a function given these conditions.