C x The slope of the tangent to the curve at every point on this curve is decreasing.
The actual values for the slopes are obtained by differentiation: y = 8x − x 2 dy
___ dx = 8 − 2x = m
x 012345678 m 86420 –2
–4 m
2 4 6 8
–2
–4 –6 –8
(7, –6) = (x2, m2)
The slope m is decreasing for all values of x ⇒ dm ___
___ dx =
dm
___ dx 2 < 0 for all points on this curve.
d 2 y
There are many points on this curve but there is only one point at which the curve fl attens out. This point is the top of the hill (point B). It is known as a local maximum point of the curve. At B, the slope of the tangent is 0 as the curve fl attens out. At a local maximum point of a curve, two conditions hold:
1. 2.
___ dx = 0 at the point [slope = 0] d 2 y
dy
___ dx 2 < 0 at the point
[slope m is decreasing from m (+) to m (–)] m +
m – dx < 0 for all x ∈ R. (2, 4) = (x1, m1) 0 12 3 4 5 6 7 8 x
