Let g(x)=-(f(-1))/(2) x^2(x-1)-f(0)(x^2-1) +(f(1))/(2) x^2(x+1)-f^'(0) x(x-1)(x+1) where f is a

step 1 So in this question we have g of x is equal to f of minus 1 by 2 x squared x minus 1 minus f of

Let g(x)=-(f(-1))/(2) x^2(x-1)-f(0)(x^2-1) +(f(1))/(2)...

Let $g(x)=-\frac{f(-1)}{2} x^{2}(x-1)-f(0)\left(x^{2}-1\right)$ $+\frac{f(1)}{2} x^{2}(x+1)-f^{\prime}(0) x(x-1)(x+1)$ where $f$ is a thrice differentiable function. Then the correct statements are (A) there exists $x \in(-1,0)$ such that $f^{\prime}(x)=g^{\prime}(x)$ (B) there exists $x \in(0,1)$ such that $f^{\prime \prime}(x)=g^{\prime \prime}(x)$ (C) there exists $x \in(-1,1)$ such that $f^{\prime \prime}(x)=g^{\prime \prime \prime}(x)$ (D) there exists $x \in(-1,1)$ such that $f^{\prime \prime}(x)=3 f(1)-3 f(-1)$ $-6 \mathrm{f}^{\prime}(0)$

Let $g(x)=-\frac{f(-1)}{2} x^{2}(x-1)-f(0)\left(x^{2}-1\right)$
$+\frac{f(1)}{2} x^{2}(x+1)-f^{\prime}(0) x(x-1)(x+1)$ where $f$ is a thrice
differentiable function. Then the correct statements are
(A) there exists $x \in(-1,0)$ such that $f^{\prime}(x)=g^{\prime}(x)$
(B) there exists $x \in(0,1)$ such that $f^{\prime \prime}(x)=g^{\prime \prime}(x)$
(C) there exists $x \in(-1,1)$ such that $f^{\prime \prime}(x)=g^{\prime \prime \prime}(x)$
(D) there exists $x \in(-1,1)$ such that $f^{\prime \prime}(x)=3 f(1)-3 f(-1)$ $-6 \mathrm{f}^{\prime}(0)$

Key Concepts

Darboux's Theorem (Intermediate Value Property of Derivatives)

Darboux's Theorem establishes that derivatives, even if they are not continuous, satisfy the intermediate value property. That is, they take on every value between any two of their values over an interval. This property is crucial for proving the existence of certain derivative values within an interval and underpins many arguments related to the behavior of function derivatives.

Mean Value Theorem

The Mean Value Theorem states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists a point where the derivative equals the average rate of change of the function over that interval. This theorem is essential in establishing the existence of certain derivative values and is frequently applied to connect discrete data points with derivative information.

Rolle's Theorem

A special case of the Mean Value Theorem, Rolle's Theorem asserts that if a differentiable function has equal values at two distinct points, then there must be at least one point between them where the derivative is zero. This theorem is often used in proofs and arguments involving the existence of stationary points in a function.

Polynomial Interpolation

This concept involves constructing a polynomial that exactly matches a function's values at a discrete set of points. It is a fundamental tool in numerical analysis and approximation theory because it allows one to estimate or simulate a complex function using a simpler polynomial expression that passes through the given data points.

Hermite Interpolation

Hermite interpolation extends polynomial interpolation by not only matching the function values at specific points but also one or more of its derivative values at those points. This provides a higher-fidelity approximation of the function’s behavior, especially when derivative information is available, leading to better error control and smoother approximations.

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