I recently took a test that asked me, "If the slope follows the equation dy/dx = y - 2x, then at the point (0,0), is there a local extreme value? Justify analytically." There was a slope field that showed how the curve would have been concave down had it passed through the point (0,0). How would I go about proving it analytically aside from saying that the slope is zero? Do I need to use integration, and if I do, how do I do that?
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Use the Second Derivative test.
d²y/dx² = dy/dx - 2
At (0, 0), the second derivative is -2, an indication that (0, 0) is a relative max.
The equation can be solved analytically, but I suspect you have not had differential equations yet. The solution of the differential equation subject to the condition y(0) = 0 is y(x) = 2(1 + x - e^x), and this function does indeed have an extreme value at x = 0. No other solution of the differential equation passes through (0,0).
d²y/dx² = dy/dx - 2
At (0, 0), the second derivative is -2, an indication that (0, 0) is a relative max.
The equation can be solved analytically, but I suspect you have not had differential equations yet. The solution of the differential equation subject to the condition y(0) = 0 is y(x) = 2(1 + x - e^x), and this function does indeed have an extreme value at x = 0. No other solution of the differential equation passes through (0,0).