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Understanding intervals of increase and decrease means identifying where a function's output becomes larger or smaller as the input (x-value) changes. To find these intervals, we use the derivative of the function. The derivative tells us the rate of change of the function.To determine these intervals for a function like \(f(x) = 2 + 3x - x^3\), we start by finding the first derivative, which is \(f'(x) = 3 - 3x^2\). This equation helps us find critical points—where the derivative equals zero or is undefined.After setting \(f'(x) = 0\), we solve for \(x\) to find the critical points. In this case, the critical points come out to be \(x = -1\) and \(x = 1\). These points divide the number line into different intervals: \((-\infty, -1)\), \((-1, 1)\), and \((1, \infty)\).Next, we examine each interval to determine the sign of \(f'(x)\) within it:- For \(x < -1\), \(f'(x)\) is negative, which means the function is decreasing.- Between \(x = -1\) and \(x = 1\), \(f'(x)\) is positive, indicating the function is increasing.- For \(x > 1\), \(f'(x)\) is negative, so the function is again decreasing.This analysis helps us conclude that the function increases on \((-1,1)\) and decreases on \((-fty,-1)\) and \((1, \infty)\).

Local maximum and minimum values occur where a function's behavior changes from increasing to decreasing or vice versa. This concept helps us identify the highest or lowest points in small sections of a graph.To find these values for \(f(x) = 2 + 3x - x^3\), we rely on the critical points found earlier. At these points, we evaluate the sign changes of the first derivative \(f'(x)\) to determine if they are local maxima or minima.- At \(x = -1\), the derivative changes from negative (as \(x\) approaches from the left) to positive (as \(x\) moves to the right). This sign change indicates a local minimum.- At \(x = 1\), the derivative changes from positive to negative. This change signifies a local maximum.For both \(x = -1\) and \(x = 1\), we calculate the function values:- \(f(-1) = 4\) is the local minimum value.- \(f(1) = 4\) represents the local maximum value.Identifying these values helps in understanding the behavior of the function at specific points along its graph.

The concavity of a function indicates the direction in which it bends or curves. When a function is concave up, it resembles a cup and curving upwards, while concave down looks like a cap and curves downwards. Inflection points are spots on a graph where the concavity changes directions.To explore concavity for the function \(f(x) = 2 + 3x - x^3\), we calculate the second derivative: \(f''(x) = -6x\). This derivative guides us to find where concavity switches.By setting \(f''(x) = 0\), we solve for \(x = 0\), revealing a potential inflection point. We then check the concavity in the intervals \((-\infty, 0)\) and \((0, \infty)\):- For \(x < 0\), \(f''(x) > 0\), showing the function is concave up.- For \(x > 0\), \(f''(x) < 0\), indicating the function is concave down.At \(x = 0\), since the concavity changes from up to down, this point is confirmed as an inflection point. Understanding these concepts allows us to better visualize and sketch the graph's overall shape.