91Ó°ÊÓ

To find the first derivative of g(x), we will use the given function \(\displaystyle g(x)=-\dfrac{f(-1)}{2} x^{2}(x-1)-f(0)\left(x^{2}-1\right)+ \dfrac{f(1)}{2} x^{2}(x+1)-f^{\prime}(0) x(x-1)(x+1)\). Using the product and chain rule, we get: $\displaystyle g^{\prime}(x)= -\dfrac{f(-1)}{2} \left(3x^{2}-2x\right) -f(0) \left(2x\right) +\dfrac{f(1)}{2} \left(3x^{2}+2x\right) -f^{\prime}(0) \left((x-1)(x+1)+x(x+1)+x(x-1)\right)$

Simplify the expression of \(g^{\prime}(x)\) by combining like terms: $\displaystyle g^{\prime}(x)= -\dfrac{f(-1)}{2} (3x^{2}-2x) -2xf(0) +\dfrac{f(1)}{2} (3x^{2}+2x) -2xf^{\prime}(0) (x^{2}-1)$

To find the second derivative of g(x), we will differentiate \(g^{\prime}(x)\) with respect to x: $\displaystyle g^{\prime\prime}(x)= -\dfrac{f(-1)}{2} (6x-2) -2f(0) +\dfrac{f(1)}{2} (6x+2) -2xf^{\prime\prime}(0)$

Let's analyze the conditions for each of the given statements: (A) there exists \(x \in(-1,0)\) such that \(f^{\prime}(x)=g^{\prime}(x)\) By comparing the expressions of \(f^{\prime}(x)\) and \(g^{\prime}(x)\), we can see that this condition is true. (B) there exists \(x \in(0,1)\) such that \(f^{\prime\prime}(x)=g^{\prime\prime}(x)\) By comparing the expressions of \(f^{\prime\prime}(x)\) and \(g^{\prime\prime}(x)\), we can see that this condition is true. (C) there exists \(x \in(-1,1)\) such that \(f^{\prime\prime}(x)=g^{\prime\prime\prime}(x)\) We haven't found \(g^{\prime\prime\prime}(x)\), but it doesn't matter for our analysis. Since f is a thrice differentiable function, it means that \(f^{\prime\prime\prime}(x)\) exists and is continuous. We can apply the Intermediate Value Theorem (IVT) on the interval \((-1,1)\), then there must exist a point \(x\) such that \(f^{\prime\prime}(x)=g^{\prime\prime\prime}(x)\). (D) there exists \(x \in(-1,1)\) such that \(f^{\prime\prime}(x)=3 f(1)-3 f(-1)-6 \mathrm{f}^{\prime}(0)\) We don't have enough information about f(x) to determine if this statement is true. So, statements (A), (B), and (C) are correct, but we cannot confirm the correctness of statement (D).