Let f be differentiable for all x. If f(1)=-2 and f^'(x) ≥2 for x ∈[1,6], then [2005] (A) f(6)

step 1 We are given that f of 1 is equal to minus 2 and f -daz x is greater than equal to 2 4 x belongs

Let f be differentiable for all x. If f(1)=-2 and f^'(x) ≥2...

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Key Concepts

Differentiability and the Derivative

Differentiability of a function implies it has a derivative, which represents the instantaneous rate of change of the function. This concept is fundamental in calculus, as it allows one to infer local behavior of functions and derive properties such as increasing or decreasing trends over an interval.

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 instantaneous rate of change (the derivative) is equal to the average rate of change over the interval. This theorem is essential for translating local information (derivatives) into global information (function values over intervals).

Rate of Change Constraints and Function Behavior

When the derivative of a function is bounded by a certain value over an interval, it provides a means to establish minimum or maximum changes in the function’s values over that interval. If the derivative is bounded below by a constant, then the function is guaranteed to increase at least at the rate dictated by that constant, allowing one to find lower bounds for the function's values at points beyond the initial condition.

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