AP Calculus AB : 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema- Exam Style questions with Answer- FRQ

Question:

The figure above shows the graph of the piecewise-linear function f. For -4 ≤ x ≤ 12, the function g is defined by\(g(x)=\int_{2}^{x}f(t)dt.\)

(a) Does g have a relative minimum, a relative maximum, or neither at x = 10 ? Justify your answer.
(b) Does the graph of g have a point of inflection at x = 4 ? Justify your answer.
(c) Find the absolute minimum value and the absolute maximum value of g on the interval -4 ≤ x ≤ 12. Justify your answers.
(d) For -4 ≤ x ≤ 12 , find all intervals for which g (x) ≤ 0.

▶️Answer/Explanation

Ans:

(a)

g'(x) = f(x)

g does not have a relative minimum  or  maximum at x = 10 because g'(x) = f(x) does not change sign at this point.

(b)

g”(x) = f'(x)

f'(x) = g”(x) does change sign at x = 4 so g does have a point of inflection at this point

(c)

g'(x) = f(x) = 0           x = -2          x = 2

                                      x = 6          x = 10

does not change sign at x = 2 and x = 10

The absolute minimum value of g on the interval -4 ≤ x ≤ 12 is -8 and the absolute maximum value of g is 8.

(d)

\(g(x)=\int_{2}^{x}f(t)dt\leq 0\)

g(x) = 0    at x = 2    and x = 10

g(x) ≤ 0 in the intervals   -4 ≤ x ≤ 2  and 10 ≤ x ≤ 12

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