Problem 1
Question
Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) has two derivatives, with \(f(0)=f^{\prime}(0)=0\) and \(\left|f^{\prime \prime}(x)\right| \leq 1\) if \(|x| \leq 1\). Prove that \(f(x) \leq 1 / 2\) if \(|x| \leq 1\).
Step-by-Step Solution
Verified Answer
Using Taylor's Theorem and the bounds on the second derivative, it is shown that \(|f(x)| \leq \frac{1}{2}\) for \(|x| \leq 1\), which implies that \(f(x) \leq \frac{1}{2}\).
1Step 1: Examine Given Information
Analyse the given information about the function f. We are told that the function is twice differentiable, which implies that it has a continuous derivative (meaning the function is smooth). Moreover, we have initial conditions: \(f(0)=f'(0)=0\) and a bound on the second derivative: \(|f''(x)| \leq 1\) for \(|x| \leq 1\).
2Step 2: Apply Taylor's Theorem
We apply Taylor's Theorem to express \(f(x)\) around the point \(x=0\). According to Taylor's Theorem for some \(c\) between \(0\) and \(x\): \[ f(x) = f(0) + f'(0)x + \frac{f''(c)}{2}x^2 \]. Since \(f(0)=f'(0)=0\), this simplifies to \[ f(x) = \frac{f''(c)}{2}x^2 \].
3Step 3: Apply the Bound on the Second Derivative
Since \(|f''(c)| \leq 1\) for any \(c\) where \(|c| \leq 1\), and \(c\) lies between 0 and \(x\), we know that \(c\) must also satisfy this inequality. Therefore, we have \[ |f''(c)| \leq 1 \Rightarrow \left|\frac{f''(c)}{2}x^2\right| \leq \frac{1}{2}x^2 \].
4Step 4: Conclude the Proof
Since \(x\) is in the interval \([-1,1]\), the maximum value of \(x^2\) is 1. Therefore, \[ \left|\frac{f''(c)}{2}x^2\right| \leq \frac{1}{2} \]. As a result, the absolute value of \(f(x)\) is at most \(\frac{1}{2}\), which implies \(f(x) \leq \frac{1}{2}\) for \(|x| \leq 1\), completing the proof.
Key Concepts
Differentiable FunctionsSecond DerivativeMathematical Proof
Differentiable Functions
Understanding the concept of differentiable functions is essential in calculus and many aspects of mathematical analysis. A differentiable function is one that has a derivative at each point in its domain. In simpler terms, if you can draw a tangent line at any point along the curve of the function, it is considered differentiable at that point.
This property is a sign of smoothness of the function's graph. For instance, a function like \(f(x) = x^2\) is differentiable everywhere because you can calculate its slope at every point. However, a function like \(f(x) = |x|\), which has a sharp corner at \(x = 0\), is not differentiable at that point.
In the context of the exercise provided, the function \(f\) has been noted to be twice differentiable over the range of \(x\) where \(\left|x\right| \leq 1\). This information will be crucial when applying Taylor's Theorem, as we'll explore next.
This property is a sign of smoothness of the function's graph. For instance, a function like \(f(x) = x^2\) is differentiable everywhere because you can calculate its slope at every point. However, a function like \(f(x) = |x|\), which has a sharp corner at \(x = 0\), is not differentiable at that point.
In the context of the exercise provided, the function \(f\) has been noted to be twice differentiable over the range of \(x\) where \(\left|x\right| \leq 1\). This information will be crucial when applying Taylor's Theorem, as we'll explore next.
Second Derivative
The second derivative plays a vital role in understanding the curvature and concavity of a function's graph. While the first derivative provides information about the function's slope or rate of change, the second derivative describes how the slope changes. In practical terms, if the second derivative is positive, the function is curving upwards, and if it is negative, the function is curving downwards.
A classic physics example is motion: the first derivative of position with respect to time is velocity, and the second derivative is acceleration, which tells us how velocity is changing. In the provided exercise, the second derivative of \(f\), denoted \(f''(x)\), is constrained by \(\left|f''(x)\right| \leq 1\) within the interval \(\left|x\right| \leq 1\). This bounding of \(f''\) restricts the steepness of the function's curvature and directly influences the maximum value that \(f(x)\) can take, as per the exercise's solution.
A classic physics example is motion: the first derivative of position with respect to time is velocity, and the second derivative is acceleration, which tells us how velocity is changing. In the provided exercise, the second derivative of \(f\), denoted \(f''(x)\), is constrained by \(\left|f''(x)\right| \leq 1\) within the interval \(\left|x\right| \leq 1\). This bounding of \(f''\) restricts the steepness of the function's curvature and directly influences the maximum value that \(f(x)\) can take, as per the exercise's solution.
Mathematical Proof
A mathematical proof is a logical argument that establishes the truth of a mathematical statement beyond any doubt. Proofs are the foundation of mathematics, cementing relationships and properties rigorously. They rely on previously established theorems, definitions, axioms, and logical reasoning to demonstrate the validity of a claim.
The exercise presented required us to prove that \(f(x) \leq 1 / 2\) under certain conditions. To achieve this, the step-by-step solution carefully pieced together information about \(f\), applied Taylor's Theorem, and used the constraints given for the second derivative. This systematic approach illustrates how proofs are constructed: by breaking down complex problems into simpler steps and using known results to reach a logical conclusion. Understanding how to structure and write proofs is an invaluable skill for students, enabling them to tackle a wide range of mathematical challenges.
The exercise presented required us to prove that \(f(x) \leq 1 / 2\) under certain conditions. To achieve this, the step-by-step solution carefully pieced together information about \(f\), applied Taylor's Theorem, and used the constraints given for the second derivative. This systematic approach illustrates how proofs are constructed: by breaking down complex problems into simpler steps and using known results to reach a logical conclusion. Understanding how to structure and write proofs is an invaluable skill for students, enabling them to tackle a wide range of mathematical challenges.
Other exercises in this chapter
Problem 1
Suppose that the functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) are differentiable and define \(h \equiv f \cir
View solution Problem 1
For each of the following statements, determine whether it is true or false and justify your answer. a. If the function \(f: \mathbb{R} \rightarrow \mathbb{R}\)
View solution Problem 2
Give a reasonable interpretation of the formula $$ \frac{d f}{d r}=\frac{d f}{d u} \cdot \frac{d u}{d s} \cdot \frac{d s}{d r} $$
View solution Problem 2
Define \(f(x)=x^{3}+2 x+1\) for all \(x .\) Find the equation of the tangent line to the graph of \(f: \mathbb{R} \rightarrow \mathbb{R}\) at the point (2,13).
View solution