6. Graphical Applications of Derivatives

The First Derivative Test

6. Graphical Applications of Derivatives

The First Derivative Test

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Multiple Choice

Identify the open intervals on which the function is increasing or decreasing.
$f(x)=3x^4+8x^3-18x^2+7$

Increasing on $\left(-\infty,-3\right)\&\left(0,1\right)$ , Decreasing on $\left(-3,0\right)\&\left(1,\infty\right)$

Increasing on $\left(-3,0\right)\&\left(1,\infty\right)$ , Decreasing on $\left(-\infty,-3\right)\&\left(0,1\right)$

Increasing on $\left(-\infty,0\right)$ , Decreasing on $\left(0,\infty\right)$

Increasing on $\left(0,\infty\right)$ , Decreasing on $\left(-\infty,0\right)$

Verified step by step guidance

Step 1: To determine where the function is increasing or decreasing, start by finding the derivative of the function, f(x). The derivative, f'(x), represents the slope of the tangent line to the curve, which helps identify intervals of increase or decrease. For f(x) = 3x^4 + 8x^3 - 18x^2 + 7, compute f'(x) using the power rule: f'(x) = 12x^3 + 24x^2 - 36x.

Step 2: Set the derivative f'(x) equal to zero to find the critical points. Solve the equation 12x^3 + 24x^2 - 36x = 0. Factorize the derivative to simplify: 12x(x^2 + 2x - 3) = 0. Further factorize the quadratic term: 12x(x + 3)(x - 1) = 0. The critical points are x = 0, x = -3, and x = 1.

Step 3: Use the critical points to divide the number line into intervals: (-∞, -3), (-3, 0), (0, 1), and (1, ∞). Test the sign of f'(x) in each interval by choosing a test point within each interval and substituting it into f'(x). For example, choose x = -4 for (-∞, -3), x = -1 for (-3, 0), x = 0.5 for (0, 1), and x = 2 for (1, ∞).

Step 4: Analyze the sign of f'(x) in each interval. If f'(x) > 0, the function is increasing on that interval. If f'(x) < 0, the function is decreasing on that interval. For example, substituting x = -4 into f'(x) gives a negative value, indicating the function is decreasing on (-∞, -3). Repeat this process for all intervals.

Step 5: Based on the sign analysis, determine the intervals of increase and decrease. Summarize the results: The function is increasing on (-3, 0) and (1, ∞), and decreasing on (-∞, -3) and (0, 1).