(a) g(- 4) ≈ - 20.566, g(0) = - 8, g(4) = 4, g(6) = 0, g(8) = - 4
(b) g(x) is increasing in the interval [- 4, 2] and decreasing in the interval [4, 8].
(c) There is an up concavity and a decreasing behavior in the interval [2, 6].
(d) The points x = 2 and x = 6 are points of inflection of g(x).
(e) The equation of the line tangent to g(x) at x = 6 is y = - 4 · x + 24.
(f) The range of g(x) is [- 20.566, 4].
(g) The graph of g(x) is shown in the picture attached below.
How to analyze the integral of a piecewise defined function
In this problem we have a piecewise defined function formed by four functions, a circle-like function and three lines, whose integral has to be analyzed in all its characteristics. (a) The integral is described graphically by the area below the curve, where g(2) = 0 and the following properties of the integral are used:
g(- 4) = g(2) - [F(2) - F(- 4)]
g(- 4) = 0 - 0.25π · 4² - 4 · 2
g(- 4) ≈ - 20.566
g(0) = g(2) - [F(2) - F(0)]
g(0) = 0 - 4 · 2
g(0) = - 8
g(4) = g(2) + [F(4) - F(2)]
g(4) = 0 + 0.5 · (2) · (4)
g(4) = 4
g(6) = g(2) + [F(6) - F(2)]
g(6) = 0 + 0.5 · (2) · (4) - 0.5 · (2) · (4)
g(6) = 0
g(8) = g(2) + [F(8) - F(2)]
g(8) = 0 + 0.5 · (2) · (4) - (2) · (4)
g(8) = - 4
(b) An interval of g(x) is increasing when f(x) > 0 and decreasing when f(x) < 0. Thus, g(x) is increasing in the interval [- 4, 2] and decreasing in the interval [4, 8].
(c) There is an up concavity and a decreasing behavior in the interval [2, 6].
(d) There are points of inflection for values of x such that f'(x) do not exists. The points x = 2 and x = 6 are points of inflection of g(x).
(e) We need to determine the slope and the intercept of the tangent line to determine the equation of the line:
Slope
m = f(6)
m = - 4
Intercept (x = 6, g(x) = 0)
b = g(x) - m · x
b = 0 - (- 4) · 6
b = 24
The equation of the line tangent to g(x) at x = 6 is y = - 4 · x + 24.
(f) The range of g(x) corresponds to the set of values of y that exists in the function. In accordance with the information given in (a), the range of g(x) is [- 20.566, 4].
(g) The graph of g(x) is shown in the picture attached below.
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