The largest interval where the function f(x) is increasing is (-1,1).
To find the intervals where the function f(x) = -x^3 + 4x + 3 is increasing or decreasing, we need to analyze its derivative, f'(x) = -3x^2 + 4
First, we find the critical points of f(x) by setting f'(x) = 0 and solving the resulting equation:
-3x^2 + 4 = 0
x^2 = 4/3
x = ± √(4/3)
The critical points are x = -√(4/3) and x = √(4/3).
Next, we divide the number line into three intervals:
Interval x-values
(-\infty, -√(4/3)) x < -√(4/3)
(-√(4/3), √(4/3)) -√(4/3) ≤ x ≤ √(4/3)
(√(4/3), ∞) x > √(4/3)
For each interval, we evaluate f'(x) at a sample point to determine whether it's positive or negative on that interval.
Interval Sample point f'(x) Increasing/Decreasing
(-\infty, -√(4/3)) x = -2 f'(-2) = 20 > 0 Increasing
(-√(4/3), √(4/3)) x = 0 f'(0) = 4 > 0 Increasing
(√(4/3), ∞) x = 2 f'(2) = -20 < 0 Decreasing
Therefore, the largest interval where the function f(x) is increasing is (-1,1).
Complete question:
Using only the values given in the table for the function f(x) = –x3 + 4x + 3, what is the largest interval of x-values where the function is increasing?