Answer:
The interval over which the function is positive is (-1 , 3)
The interval over which the function is negative is (-∞ , -1)∪(3 , ∞)
Step-by-step explanation:
* Lets explain the meaning of The intervals of positive and negative
values of a function
- Interval of positive values means the values of x when the values
of y are positive
- Y is positive means the graph is above the x-axis
- Interval of negative values means the values of x when the values
of y are negative
- Y is negative means the graph is below the x-axis
- <u><em>Remember</em></u> that the value of y on the x-axis is zero, then the points
of intersection between the graph and x-axis have zero value of y
and zero is not positive nor negative, then the x-coordinate of
these points do not belong to the intervals of positive or negative
* <em>Lets solve the problem</em>
∵ The graph represents a downward parabola
∵ The positive value of the parabola is above the x-axis
∵ The parabola intersects the x-axis at -1 , 3
∴ Y is positive at ⇒ -1 < x < 3
# -1 , 3 ∉ to the interval
∴ The interval over which the function is positive is (-1 , 3)
∵ The negative value of the parabola is below the x-axis
∵ The parabola intersects the x-axis at -1 , 3
∴ Y is negative at ⇒ -∞ < x < -1 and 3 < x < ∞
# -1 , 3 ∉ to the interval
∴ The interval over which the function is negative is (-∞ , -1)∪(3 , ∞)
* <u><em>The error of the student he wrote Positive</em></u>: [-1 , 3] which means
that -1 , 3 ∈ to the intervals over positive
which the function is positive or negative
, and the integrand 
comes from integrating with respect to 
.