4. Consider the curve given by the equation
2( x − y )= 3 + cos y . For all points on the
curve,
2
3
≤
dy
dx ≤ 2 .
(a) Show that dy
dx =
2
2 − sin y
.
(b) For −
π
2
< y <
π
2
, there is a point P on
the curve through which the line tangent to the
curve has slope 1 . Find the coordinates of the
point P .
(c) Determine the concavity of the curve at points for
which −
π
2
< y <
π
2
. Give a reason for your
answer.
(d) Let y = f ( x) be a function, defined implicitly
by 2( x − y )= 3 + cos y , that is continuous on
the closed interval [2, 2.1] and differentiable
on the open interval (2, 2.1) . Use the Mean
Value Theorem on the interval [2, 2.1] to show
that 1
15 ≤ f (2.1) − f (2) ≤
1
5