The figure above shows the graph of g(x), where g is the derivative of the function f, for −3 £ x
£ 9. The graph consists of three semicircular regions and has horizontal tangent lines at x = 0, x
= 4.5, and x = 7.5.
(a) Find all values of x, for −3 < x £ 9, at which f attains a relative minimum. Justify your
answer.
Because g is the derivative of the function f, f will attain a relative minimum at a point where g
= 0 and where g is negative to the left of that point and positive to the right of it. This occurs at
x = 6.
(b) Find all values of x, for −3 < x £ 9, at which f attains a relative maximum. Justify your
answer.
Because g is the derivative of the function f, f will attain a relative maximum at a point where g
= 0 and where g is positive to the left of that point and negative to the right of it. This occurs at
x = 3.
(c) If f(x) = g(t) dt, find f (6).
We are trying to find the area between the graph and the x-axis from x = −3 to x = 6. From x =
−3 to x = 3, the region is a semicircle of radius 3, so the area is .
From x = 3 to x = 6, the region is a semicircle of radius , so the area is . We subtract the
latter region from the former to obtain: .
(d) Find all points where f′′(x) = 0.
Because f′′(x) = g’(x), we are looking for points were the derivative of g is zero. This occurs
at the horizontal tangent lines at x = 0, x = 4.5, and x = 7.5.
- Consider the curve given by x^2 y − 4x + y^2 = 2.
(a) Find .
We can find by implicit differentiation: x^2 + 2xy − 4 + 2y = 0.
