If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Worked example: Inflection points from first derivative

## Video transcript

so we're told let G be a differentiable function defined over the closed interval from negative six to six the graph of its derivative so they're giving the graphing the derivative of G G prime is given below so this isn't the graph of G this is the graph of G prime what is the x value of the leftmost inflection point inflection point in the graph of G so they want they don't want to know the the x value of the inflection points in the graph of G prime and the graph in this graph they want to know the inflection points the X values of the inflection points in the graph of G and we have to figure out the leftmost one so let me just make a little table here to think about what is happening at inflection points in our second derivative our first derivative and our actual function so this is G prime prime this is G prime and this is our actual I call it the original function so an inflection point are points where our second derivative is switching signs it's going from positive negative or negative to positive so let's consider that first scenario so G so going from positive to negative positive positive to negative so if G prime prime if the second derivative is going from positive and negative what is the first derivative to be doing well remember the second derivative is the derivative of the first derivative so where the second derivative is positive where the second derivative is positive that means that the first derivative is increasing so if second derivative is going from positive to negative that means first derivative is going from increasing to decreasing from increasing to decreasing and the function itself well when the second derivative is positive we are going to be that means that means that the slope is constantly increasing and so that means we are concave upwards so concave upwards upwards to downwards to concave to concave downwards but they've given us the graph of G prime so let's focus one of the points where G prime is going from increasing to decreasing so let's see G prime is increasing increasing increasing increasing increasing at a slower rate then it starts decreasing so right over there it's going from increasing to decreasing so then it's decreasing decreasing decreasing then it goes increasing increasing increasing increasing and then decreasing again so that's another point where we're going from increasing to decreasing and those are the only ones that look like we're going from increasing to decreasing but we're not done yet because it's not just about going from the second derivative going from positive to negative it's also the other way around any time the second derivative is switching signs so it's also the situation where we're going from negative to positive or for the first derivative is going from decreasing decreasing to increasing decreasing to increasing well let's see we are decreasing decreasing decreasing and then we increasing all right so it's right there and then we're increasing decreasing decreasing decreasing and then we're increasing so right over there so these are the inflection points that I've just figured out visually so if you look at the choices if we want to answer the original question well the leftmost one is that X is equal to negative 3 it's x equals negative 3 x equals negative 1 is indeed a x value where we have an inflection point and let's see x equals 2 is 1 and so is x equals 4 so they actually listed all of these are inflection points and they just wanted the leftmost one