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Determine the points where the graph of the function f(x) = 2x^3 - 6x^2 + 2x - 4 is concave down.

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For a function f(x), to determine if it is concave up or concave down at a point where x = c, we have to first determine whether the second derivative f''(c) exists. If function f(x) has a second derivative at x = c and f''(c)<0, the function is concave down at that point.

Here f(x) = 2x^3 - 6x^2 + 2x - 4

f'(x) = 6x^2 - 12x + 2

f''(x) = 12x - 12

= 12(x - 1)

To determine the points where the function is concave down, solve 12(x - 1)<0

=> x - 1 < 0

=> x < 1

This shows that for all points where x < 1, the function is concave down.


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