Answer:
a) The absolute maximum is 401 and the absolute minimum is 9.
b) The absolute maximum is 349 and the absolute minimum is -99.
c) The absolute maximum is 401 and the absolute minimum is -99.
Step-by-step explanation:
The absolute minimum and absolute maximum values are determined with the help of the First and Second Derivative Tests:
FDT
[tex]3\cdot x^{2} + 12\cdot x - 63 = 0[/tex]
The roots of the function are: [tex]x_{1} = 3[/tex] and [tex]x_{2} = -7[/tex]. Each point is evaluated in the second derivative of the function:
SDT
[tex]f''(x) = 6\cdot x + 12[/tex]
[tex]f''(x_{1}) = 30[/tex] (Absolute minimum)
[tex]f''(x_{2}) = -30[/tex] (Absolute maximum)
The values for each extreme are, respectively:
[tex]f(x_{1}) = -99[/tex]
[tex]f(x_{2}) = 401[/tex]
Now, each interval is analyzed herein:
a) The absolute maximum is 401 and the absolute minimum is 9.
b) The absolute maximum is 349 and the absolute minimum is -99.
c) The absolute maximum is 401 and the absolute minimum is -99.
