calculator allowed\n4.\nlet ( f ( x ) = int _ { a } ^ { x } h ( t ) d t ), where ( h ) has the graph shown…
calculator allowed\n4.\nlet ( f ( x ) = int _ { a } ^ { x } h ( t ) d t ), where ( h ) has the graph shown above. which of the following could be the graph of ( f )?\n(a)\n(b)\n(c)\n(d)\n(e)
Answer
Explanation:
Step1: Use the Fundamental Theorem of Calculus
By the Fundamental Theorem of Calculus, (f^{\prime}(x)=h(x)). So the derivative of (f(x)) is given by the graph of (h(x)).
Step2: Analyze the sign of (h(x))
- When (x < b) (except at critical points), (h(x)) is positive for (a < x< b) (before the (x -)intercept of (h(x)) at (x = b)), so (f(x)) is increasing on ((a,b)) (since (f^{\prime}(x)=h(x)>0) on ((a,b))).
- When (x>b) (in a neighborhood of (x = b)), (h(x)) is negative, so (f(x)) is decreasing on ((b,c)) (since (f^{\prime}(x)=h(x)<0) on ((b,c))).
- Also, (f(a)=\int_{a}^{a}h(t)dt = 0).
Step3: Check each option
- Option (A): (f(a)\neq0), so it is incorrect.
- Option (B): The function (f(x)) is not increasing on ((a,b)) as (h(x)>0) on ((a,b)), so it is incorrect.
- Option (C): (f(a)\neq0), so it is incorrect.
- Option (D): The function (f(x)) is not increasing on ((a,b)) as (h(x)>0) on ((a,b)), so it is incorrect.
- Option (E): (f(a) = 0), (f(x)) is increasing on ((a,b)) (because (h(x)>0) on ((a,b))) and decreasing on ((b,c)) (because (h(x)<0) on ((b,c)))
Answer:
E.