Proving the Existence of a Second Derivative within a Given Interval

Given a function f that is twice differentiable on the interval [0,4] and satisfies the conditions f'(1) 0, f'(2) 1, f'(3) 1, we aim to prove the existence of a point d in [0,4] such that the second derivative f''(d) 1/2. This proof relies heavily on the Mean Value Theorem (MVT) and the intermediate value property of derivatives.

Step-by-Step Proof

First, let's consider the function f on the interval [1,2]. By the Mean Value Theorem, there exists a point x_1 in (1,2) such that:

"f'(x_1) 0

Similarly, on the interval [2,3], by the MVT, there exists a point x_2 in (2,3) such that:

"f'(x_2) 1

Thus, we have:

x_1 u2018 x_2 and x_1

Next, by the Mean Value Theorem applied to the function g(x) f'(x) on the interval [x_1, x_2], there exists a point x u2018 [x_1, x_2] such that:

"f''(x) u2061 frac{1 - 0}{x_2 - x_1} u2061 frac{1}{x_2 - x_1}

Since x_1 u2061 frac{1}{2}

Therefore, we conclude that:

"f''(x) u2061 frac{1}{2}

This is a direct application of the Mean Value Theorem.

Further Analysis

Furthermore, if we consider the function f on the intervals [1,2] and [2,3], we can apply the MVT to the slopes between the points (1, f(1)), (2, f(2)), and (3, f(3)). By the MVT, there exist points d_1 in (1,2) and d_2 in (2,3) such that:

"f'(d_1) u2061 frac{f(2) - f(1)}{2 - 1} 0 f'(d_2) u2061 frac{f(3) - f(2)}{3 - 2} 1

Now, consider the interval [d_1, d_2]. By the MVT, there exists a point d in [d_1, d_2] such that:

"f''(d) u2061 frac{f'(d_2) - f'(d_1)}{d_2 - d_1} u2061 frac{1 - 0}{d_2 - d_1} u2061 frac{1}{d_2 - d_1}

The denominator d_2 - d_1 is smaller than 2, so:

"f''(d) u2061 frac{1}{d_2 - d_1} > u2061 frac{1}{2}

Therefore, we conclude that there exists a point d in [0,4] such that:

"f''(d) u2061 frac{1}{2}

This completes our proof using the Mean Value Theorem.

Conclusion

The proof relies on the Mean Value Theorem and the intermediate value property of derivatives. By carefully applying these principles, we have demonstrated the existence of a point within the given interval where the second derivative is exactly 1/2. This method can be generalized to solve similar problems involving the existence of derivatives within specified intervals.