Largest Value of a Function Given Its Derivative

In this article, we will explore the concept of finding the largest value of a function given its derivative and certain boundary conditions. Specifically, we will use the Mean Value Theorem and the Fundamental Theorem of Calculus (FTC) to solve a problem involving a function f(x) that is continuous and differentiable over an interval [-7, 0]. We will also utilize the inequality f'(x) ≤ 2 to find the upper bound of the function at a specific point.

Understanding the Problem

Suppose that f(x) is continuous and differentiable on the interval [-7, 0], and that f(-7) -3 and f'(x) ≤ 2. Our goal is to find the largest possible value for f(0).

Using the Derivative

The derivative f'(x) ≤ 2 indicates that the function f(x) is increasing at a rate of at most 2 units per unit increase in x. This implies that the slope of the tangent to the curve of f(x) does not exceed 2.

Applying the Mean Value Theorem (MVT)

The Mean Value Theorem (MVT) states that for a function that is continuous on [a, b] and differentiable on (a, b), there exists some c in (a, b) such that:

[ f'(c) frac{f(b) - f(a)}{b - a} ]

In our case, we have:

[ f'(c) leq frac{f(0) - f(-7)}{0 - (-7)} frac{f(0) - (-3)}{7} frac{f(0) 3}{7} leq 2 ]

Rearranging this inequality:

[ frac{f(0) 3}{7} leq 2 ] [ f(0) 3 leq 14 ] [ f(0) leq 11 ]

Hence, the largest possible value for f(0) is 11.

Using the Fundamental Theorem of Calculus (FTC)

The Fundamental Theorem of Calculus (FTC) states that if g(x) is continuous on [a, b] and F(x) is an antiderivative of g(x) on [a, b], then:

[ int_{a}^{b} g(x) , dx F(b) - F(a) ]

For our function f(x), we have:

[ f(0) - f(-7) int_{-7}^{0} f'(x) , dx ]

Given that f'(x) ≤ 2, we can write:

[ f(0) - f(-7) leq int_{-7}^{0} 2 , dx 2 cdot (0 - (-7)) 14 ]

Therefore:

[ f(0) leq f(-7) 14 -3 14 11 ]

Thus, the largest possible value for f(0) is again 11.

Example Function

Consider the example function f(x) 2x 11. Here, the derivative is f'(x) 2, and by evaluating at the endpoints:

[ f(0) 2(0) 11 11 ]

This confirms our result.

Conclusion

The largest possible value for f(0) given the conditions is 11. This is derived using both the Mean Value Theorem and the Fundamental Theorem of Calculus. The application of these theorems helps us establish the upper bound of the function within the given interval and derivative constraints.