The number of admissible values of C obtained | vedclass

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(C) Given the function $f(x) = x$ on the interval $[2, 5]$.Since $f(x)$ is a polynomial function, it is continuous on $[2, 5]$ and differentiable on $

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infinite

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(C) Given the function $f(x) = x$ on the interval $[2, 5]$.Since $f(x)$ is a polynomial function, it is continuous on $[2, 5]$ and differentiable on $(2, 5)$.According to Lagrange's Mean Value Theorem $(LMVT)$, there exists at least one $C \in (2, 5)$ such that $f'(C) = \frac{f(5) - f(2)}{5 - 2}$.Calculating the derivative: $f'(x) = 1$, so $f'(C) = 1$.Calculating the slope: $\frac{f(5) - f(2)}{5 - 2} = \frac{5 - 2}{5 - 2} = \frac{3}{3} = 1$.Thus, the condition $f'(C) = 1$ becomes $1 = 1$, which is true for all $C \in (2, 5)$.Since there are infinitely many points in the interval $(2, 5)$, there are infinitely many admissible values of $C$.

The number of admissible values of $C$ obtained when the Lagrange's mean value theorem is applied for the function $f(x)=x$ on $[2,5]$ is

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