Question

In your lab, a substance's temperature has been observed to follow the function T(x) = (x + 5)3 + 7. The turning point of the graph is where the substance changes from a liquid to a solid. Explain to your fellow scientists how to find the turning point of this function?

Answer 1

A turning point occurs when the velocity is equal to zero, but the acceleration is not equal to zero. 

t(x)=(x+5)^3+7

dt/dx=3(x+5)^2

d2t/dx2=6(x-5)

dt/dx=0 only when x=-5

However, since d2t/dx2(-5)=0, this point is an inflection point, not a turning point.

So there is no turning point for this function.

Now in this problem, it is even easier than the above to show that there is no turning point.  A turning point by definition is when the derivative or velocity changes sign.  Since in this case v=3(x+5)^2, for any value of x, v≥0, and thus never becomes negative, so it never changes from a positive to negative velocity because velocity in this instance is a squared function.