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