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The function ff is defined by f, of, x, equals, 2, x, cubed, plus, 2, x, squared, plus, 3, xf(x)=2x 3 +2x 2 +3x and the point left bracket, minus, 1, comma, minus, 3, right bracket(−1,−3) is on the graph of f, .f. If gg is the inverse function of ff, what is the value of g, prime, of, minus, 3, question markg ′ (−3)?

Question

The function ff is defined by f, of, x, equals, 2, x, cubed, plus, 2, x, squared, plus, 3, xf(x)=2x 3 +2x 2 +3x and the point left bracket, minus, 1, comma, minus, 3, right bracket(−1,−3) is on the graph of f, .f. If gg is the inverse function of ff, what is the value of g, prime, of, minus, 3, question markg ′ (−3)?

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Solution

To find the value of g′(−3), we first need to find the derivative of the function f(x) = 2x^3 + 2x^2 + 3x.

The derivative of f(x) is given by f′(x) = 6x^2 + 4x + 3.

We know that the point (−1,−3) is on the graph of f, so the point (−3,−1) will be on the graph of the inverse function g.

The derivative of the inverse function g at x = -3 is given by 1 / f′(g(−3)). Since g(−3) = -1, we substitute -1 into f′(x) to get f′(g(−3)) = 6*(-1)^2 + 4*(-1) + 3 = 6 - 4 + 3 = 5.

Therefore, g′(−3) = 1 / f′(g(−3)) = 1 / 5 = 0.2.

This problem has been solved

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