- Set the second derivative \(f''(x) = 12(x+7)^{11}(x-1)^7 = 0\) to find critical points at \(x = -7\) and \(x = 1\).
- Test the sign of \(f''(x)\) in the intervals \((-\infty, -7)\), \((-7, 1)\), and \((1, \infty)\) to check for concavity changes.
- Identify that a sign change occurs at both \(x = -7\) and \(x = 1\).
- Conclude that the inflection points are at \(\boxed{-7, 1}\).
Explanation
- Identify the critical values for the second derivative
The inflection points of a function \(f(x)\) occur where its second derivative, \(f''(x)\), changes sign, provided the function is continuous at those points. Given \(f''(x) = 12(x+7)^{11}(x-1)^7\), we first identify the critical points where \(f''(x) = 0\). These are found by setting the factors of the second derivative to zero: \(x+7 = 0 \implies x = -7\) and \(x-1 = 0 \implies x = 1\).
- Test the sign of the second derivative in the identified intervals
To determine if the concavity changes at these points, we examine the sign of \(f''(x)\) in the intervals created by the critical values: \((-\infty, -7)\), \((-7, 1)\), and \((1, \infty)\).
- For \(x \in (-\infty, -7)\), choose \(x = -8\): \(f''(-8) = 12(-8+7)^{11}(-8-1)^7 = 12(-1)^{11}(-9)^7 = 12(-1)(\text{negative}) > 0\).
- For \(x \in (-7, 1)\), choose \(x = 0\): \(f''(0) = 12(0+7)^{11}(0-1)^7 = 12(7)^{11}(-1)^7 = 12(\text{positive})(\text{negative}) < 0\).
- For \(x \in (1, \infty)\), choose \(x = 2\): \(f''(2) = 12(2+7)^{11}(2-1)^7 = 12(9)^{11}(1)^7 = 12(\text{positive})(\text{positive}) > 0\).
- Determine inflection points by evaluating sign changes
Since the sign of the second derivative changes at \(x = -7\) (from positive to negative) and at \(x = 1\) (from negative to positive), the function \(f(x)\) undergoes a change in concavity at these points. Therefore, both \(x = -7\) and \(x = 1\) are inflection points of the function.
- Final Answer
The inflection points of the function occur at \(x = -7\) and \(x = 1\). Thus, the values are \(\boxed{-7, 1}\).
Examples
Inflection points are crucial in engineering and physics, particularly when analyzing the structural integrity of beams or the motion of objects. For instance, in structural engineering, an inflection point in a beam's deflection curve signifies a location where the bending moment changes from positive to negative, indicating a transition in the stress distribution. Identifying these points allows engineers to strategically place reinforcements, ensuring the structure remains stable under load. Similarly, in kinematics, understanding when the acceleration changes sign helps in designing smoother motion profiles for mechanical systems.