Page:Calculus Made Easy.pdf/102

N.B.–For the particular value of $x$ that makes $y$ a minimum, the value of ${\dfrac {dy}{dx}}=0$.

If a curve first ascends and then descends, the values of ${\dfrac {dy}{dx}}$ will be positive at first; then zero, as the summit is reached; then negative, as the curve slopes downwards, as in Fig. 16. In this case $y$ is said to pass by a maximum, but the maximum value of $y$ is not necessarily the greatest value of $y$. In Fig. 28, the maximum of $y$ is $2{\tfrac {1}{3}}$, but this is by no means the greatest value $y$ can have at some other point of the curve.

Fig. 16.	Fig. 17.

N.B.–For the particular value of $x$ that makes $y$ a maximum, the value of ${\dfrac {dy}{dx}}=0$.

If a curve has the peculiar form of Fig. 17, the values of ${\dfrac {dy}{dx}}$ will always be positive; but there will be one particular place where the slope is least steep, where the value of ${\dfrac {dy}{dx}}$ will be a minimum; that is, less than it is at any other part of the curve.