Let f:[2,7] →[0, ∞) be a continuous and differentiable...

Let $\mathrm{f}:[2,7] \rightarrow[0, \infty)$ be a continuous and differentiable function. Then the value of $(f(7)-f(2)) \frac{(f(7))^{2}+(f(2))^{2}+f(2) \cdot(7)}{3}$, is (where $c \in(2,7))$ ). (A) $3 \mathrm{f}^{2}(\mathrm{c}) \cdot \mathrm{f}^{\prime}(\mathrm{c})$ (B) $4 \mathrm{f}^{(}(\mathrm{c}) \cdot \mathrm{f}^{\prime}(\mathrm{c})$ (C) $5 \mathrm{f}^{2}(\mathrm{c}) \cdot \mathrm{f}^{\prime}(\mathrm{c})$ (D) None of these

Let $\mathrm{f}:[2,7] \rightarrow[0, \infty)$ be a continuous and differentiable function. Then the value of $(f(7)-f(2)) \frac{(f(7))^{2}+(f(2))^{2}+f(2) \cdot(7)}{3}$, is (where $c \in(2,7))$ ).
(A) $3 \mathrm{f}^{2}(\mathrm{c}) \cdot \mathrm{f}^{\prime}(\mathrm{c})$
(B) $4 \mathrm{f}^{(}(\mathrm{c}) \cdot \mathrm{f}^{\prime}(\mathrm{c})$
(C) $5 \mathrm{f}^{2}(\mathrm{c}) \cdot \mathrm{f}^{\prime}(\mathrm{c})$
(D) None of these

Key Concepts

Mean Value Theorem

The Mean Value Theorem is a central result in differential calculus stating that for any function that is continuous on a closed interval and differentiable on the open interval, there exists at least one point in the open interval where the instantaneous rate of change (the derivative) equals the average rate of change over the interval. This theorem is often used to relate a change in the function's values at the endpoints of an interval to its derivative at some intermediate point.

Difference of Cubes Factorization

The algebraic identity for the difference of cubes, which states that a³ – b³ = (a – b)(a² + ab + b²), is an important tool in simplifying expressions and factoring polynomials. In many calculus problems this identity helps to relate differences in values of a function raised to a power to a product involving differences and sums of squares and products, which can then be connected to the derivatives of the function through the Mean Value Theorem.

Chain Rule

The Chain Rule is a fundamental principle in differentiation which is used when differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. This rule is particularly useful when considering functions raised to a power, such as finding the derivative of f³, which simplifies to 3f²f', a form that often appears when applying the Mean Value Theorem in problems involving differences of cubes.

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