(a) Show that if f is differentiable on − ∞ , + ∞ , and if y = f x and y = f ′ x are graphed in the same coordinate system , then between any two x -intercepts of f there is at least one x -intercept of f ′ . (b) Give some examples that illustrate this.
(a) Show that if f is differentiable on − ∞ , + ∞ , and if y = f x and y = f ′ x are graphed in the same coordinate system , then between any two x -intercepts of f there is at least one x -intercept of f ′ . (b) Give some examples that illustrate this.
(a) Show that if
f
is differentiable on
−
∞
,
+
∞
,
and if
y
=
f
x
and
y
=
f
′
x
are graphed in the same coordinate system, then between any two x-intercepts of
f
there is at least one x-intercept of
f
′
.
(b) Give some examples that illustrate this.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
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