GPT答：

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To find the coordinates of the local maximum, we need to find the critical points of the curve. 

First, let's take the derivative of y with respect to x:

dy/dx = 3x^2 - 42x^6 + 9

Next, we set this derivative equal to zero and solve for x:

3x^2 - 42x^6 + 9 = 0

This equation is a polynomial equation that can be difficult to solve analytically. However, we can use numerical methods or graphing software to approximate its solutions.

Using a graphing calculator or software, we find that there are two real solutions: approximately x ≈ -0.408 and x ≈ 1.224.

Now that we have these critical points, let's evaluate y at these values:

For x ≈ -0.408:
y ≈ (-0.408)^3 - 6(-0.408)^7 + 9(-0.408) ≈ -4

For x ≈ 1.224:
y ≈ (1.224)^3 - 6(1.224)^7 + 9(1.224) ≈ -4

Therefore, both critical points have a y-coordinate of approximately -4.

The coordinates of the local maximum are (-0.408,-4) and (1.224,-4).