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Math  /  Calculus

QuestionIf f(x)=∫0x(1−t2)et3dtf(x)=\int_{0}^{x}\left(1-t^{2}\right) e^{t^{3}} d t for all xx, then find the largest open interval on which ff is increasing.
Answer (in interval notation): □\square

Studdy Solution

STEP 1

1. The function f(x)=∫0x(1−t2)et3 dt f(x) = \int_{0}^{x} (1-t^2) e^{t^3} \, dt is defined for all x x .
2. To determine where f(x) f(x) is increasing, we need to find where its derivative f′(x) f'(x) is positive.
3. The Fundamental Theorem of Calculus can be used to find f′(x) f'(x) .

STEP 2

1. Apply the Fundamental Theorem of Calculus to find f′(x) f'(x) .
2. Determine where f′(x)>0 f'(x) > 0 .
3. Identify the largest open interval where f′(x)>0 f'(x) > 0 .

STEP 3

Apply the Fundamental Theorem of Calculus to find the derivative f′(x) f'(x) :
f′(x)=(1−x2)ex3 f'(x) = (1-x^2) e^{x^3}

STEP 4

Determine where f′(x)>0 f'(x) > 0 :
(1−x2)ex3>0 (1-x^2) e^{x^3} > 0
Since ex3>0 e^{x^3} > 0 for all x x , we only need to solve:
1−x2>0 1-x^2 > 0

STEP 5

Solve the inequality 1−x2>0 1-x^2 > 0 :
1>x2 1 > x^2
This implies:
−1<x<1 -1 < x < 1

STEP 6

Identify the largest open interval where f′(x)>0 f'(x) > 0 . From the inequality −1<x<1 -1 < x < 1 , the largest open interval is:
(−1,1) (-1, 1)
The largest open interval on which f f is increasing is (−1,1) \boxed{(-1, 1)} .

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