Ordering X-Values by Second Derivative
Consider the particular solution $$y=\sqrt{4x^2+25}$$ to the differential equation $$\frac{dy}{dx}=\frac{4x}{y}$$. The second derivative is $$\frac{d^2y}{dx^2}=\frac{100}{(\sqrt{4x^2+25})^3}$$. The approximate values of the second derivative for selected values of $$x$$ are as follows:
- At $$x = -3$$, $$\frac{d^2y}{dx^2} \approx 0.210$$
- At $$x = -1$$, $$\frac{d^2y}{dx^2} \approx 0.642$$
- At $$x = 1$$, $$\frac{d^2y}{dx^2} \approx 0.642$$
- At $$x = 3$$, $$\frac{d^2y}{dx^2} \approx 0.210$$
Based on this information, which of the following correctly orders the given $$x$$-values from least to greatest according to their corresponding second derivative values? If two second derivative values are equal, the smaller $$x$$-value should be listed first.
A
1,-1,3,-3
B
-3,3,-1,1
C
-1,1,-3,3
D
3,-3,-1,1
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