Hi, i got the question:
"Determine the x-coordinate of inflexion on the graph: y=x^5 - x^4"
How do i use differentiation to solve that?
thanks
Krizalid is right, and here is the reason: The second derivative is actually the derivative of the first derivative :P This means that the second derivative tells you the slope of the first derivative, so when the second derivative is positive, the slope of the first derivative is increasing, so the slope is going towards infinity (getting higher and higher), since the derivative tells you the slope of the function, and it is increasing, the slope of the function must be increasing, so it is convexed up. (you can see that if it is less than zero, it must be going towards zero, and if it is greater than zero, it must be going towards infinity)
And as the second derivative is negative, the slope of the first derivative must be going down, which means the slope of the first derivative is decreasing, so the slope of the function is decreasing, so it is convexed down. (if positive, will go to zero, if negative will go to negative infinity.)
Now, there must be a point where they go from convexed up to convexed down and vice versa, and as you can see that it is the sign of the second derivative which determines this, then you can see that when the sign changes between positive and negative is when the graph of the function changes convex. So when the second derivative = zero, that is the point where its sign changes, so that is the point where the function changes convex, which is the point of inflection.