- Calculate the first and second derivatives of the function to be \(g'(x) = -8x^3 + 24x^2 - 24x\) and \(g''(x) = -24(x - 1)^2\).
- Find the roots of the second derivative, identifying \(x = 1\) as the only potential inflection point.
- Evaluate the sign of the second derivative around \(x = 1\) to determine if it changes concavity; since the sign remains negative on both sides, the function has no point of inflection.
- Conclude that \(\boxed{\text{D}}\) is the correct choice.
Explanation
- Finding the Second Derivative
To identify the points of inflection of the function \(g(x) = -2x^4 + 8x^3 - 12x^2\), we must first find its second derivative, \(g''(x)\), and analyze where it changes sign. A point of inflection exists at \(x=c\) if \(g''(x)\) changes sign as \(x\) passes through \(c\). First, we calculate the first derivative:
\[g'(x) = \frac{d}{dx}(-2x^4 + 8x^3 - 12x^2) = -8x^3 + 24x^2 - 24x\]
Next, we calculate the second derivative:
\[g''(x) = \frac{d}{dx}(-8x^3 + 24x^2 - 24x) = -24x^2 + 48x - 24\]
- Finding Critical Values of the Second Derivative
To find potential points of inflection, we set the second derivative equal to zero:
\[-24x^2 + 48x - 24 = 0\]
We can simplify this equation by factoring out \(-24\):
\[-24(x^2 - 2x + 1) = 0\]
Recognizing the perfect square trinomial inside the parentheses, we have:
\[-24(x - 1)^2 = 0\]
Solving this, we find only one critical value: \(x = 1\).
- Testing for Concavity Change and Conclusion
Now we check if the sign of \(g''(x)\) changes at \(x = 1\).
- For values where \(x < 1\), such as \(x = 0\), the second derivative is \(g''(0) = -24(0 - 1)^2 = -24\), which is negative.
- For values where \(x > 1\), such as \(x = 2\), the second derivative is \(g''(2) = -24(2 - 1)^2 = -24\), which is also negative.
Since the sign of \(g''(x)\) remains negative on both sides of \(x = 1\), the concavity does not change. Therefore, \(x = 1\) is not a point of inflection. Having no other roots for \(g''(x)=0\), we conclude that the function has no points of inflection. The correct answer is D.
Examples
Understanding points of inflection is crucial in engineering and physics, such as when designing rollercoaster tracks or vehicle suspension systems. A point of inflection marks a transition where the curvature changes from concave upward to downward or vice versa, which is where a passenger would experience a shift in the direction of the inertial force. Identifying these points allows designers to ensure a smooth and safe transition for riders.