Derivative & the direction of a function

a function f of X is plotted below highlight an interval where f prime of X or the first derivative of F with respect to X is greater than zero so if our derivative F prime of X is greater than zero that means that the slope of the tangent line is positive that means that the function must be increasing whatever this is true so where is a function increasing so over here we see that as x increases as x increases our function is going to smaller and smaller values so our function is decreasing this entire place then it's it's decreasing at slower and slower rates all the way to the point right over here where the function the slope of the tangent line is flat and then the function starts increasing it starts increasing it faster and faster and faster rates so we could put this blue area anywhere from here to anywhere in this region here where the function itself is increasing notice the function could even be negative but the function is increasing so even this would be a valid place that would be valid anywhere where the functions rate of the when x increases the function is actually getting larger so let's check our answer and let's do a couple more of these a function f of X is plotted below highlight an interval where f of X is greater than zero so f of X should be sitting in the first or second quadrant right over here needs to be greater than zero and F prime of X is less than zero so f prime of x less than zero means that the function is decreasing the slope of the tangent line is negative so let's think about it the two areas where f of X is greater than zero they are this interval right over here and this interval right over here but we also care about the function decreasing this won't be valid because the function is increasing here it's increasing it's slower and slower rates but it is increasing the function is going up as X goes up if we go over here the function is decreasing as x increases so this seems to meet our constraints this this area right over here won't because the function is positive but it's also increasing so here the derivative is positive here the derivative is positive here the derivative is negative so the function itself is positive but the slope of the tangent line is negative its decreasing as x increases let's do one more a function f is plotted below highlight interval where f of X is greater than zero and f prime of X is less than zero so the same exact idea where are we positive but our function is actually decreasing so it's positive here but we can't pick that it's positive here but we can't pick that the function is decreasing here but the function isn't positive there the derivative is negative but the function isn't positive so we that one made it make it so this is the only region that we can kind of throw it in and we got it right again