Question

Let R be the set of real numbers and f: R → R, be a differentiable function such that |f(x) - f(y)| ≤ |x - y|³ ∀x, y∈ R. If f(10) = 100, then the value of f(20) is equal to:

• A. 0
• B. 10
• C. 20
• D. 100

Answer 1

Answer: (D)

100

Answer 2

The problem asks us to find the value of $f(20)$ given a differentiable function $f: R \to R$ with the property $|f(x) - f(y)| \le |x - y|^3$ for all $x, y \in R$, and $f(10) = 100$.

Explanation of the solution:

1. Analyze the given inequality: We are given the condition $|f(x) - f(y)| \le |x - y|^3$ for all $x, y \in R$.

2. Form the difference quotient: To relate this inequality to the derivative, we consider the difference quotient. For any $x \neq y$, we can divide both sides by $|x - y|$: $\frac{|f(x) - f(y)|}{|x - y|} \le \frac{|x - y|^3}{|x - y|}$ This simplifies to: $\left|\frac{f(x) - f(y)}{x - y}\right| \le |x - y|^2$ Since the absolute value is always non-negative, we also have $0 \le \left|\frac{f(x) - f(y)}{x - y}\right|$. So, we have: $0 \le \left|\frac{f(x) - f(y)}{x - y}\right| \le |x - y|^2$

3. Apply limits to find the derivative: The function $f$ is stated to be differentiable, which means $f'(x) = \lim_{y \to x} \frac{f(y) - f(x)}{y - x}$ exists for all $x \in R$. Now, we take the limit as $y \to x$ for all parts of the inequality: $\lim_{y \to x} 0 \le \lim_{y \to x} \left|\frac{f(y) - f(x)}{y - x}\right| \le \lim_{y \to x} |y - x|^2$ Evaluating the limits:

   • $\lim_{y \to x} 0 = 0$
   • $\lim_{y \to x} \left|\frac{f(y) - f(x)}{y - x}\right| = |f'(x)|$ (since the limit of the absolute value is the absolute value of the limit, provided the limit exists)
   • $\lim_{y \to x} |y - x|^2 = |x - x|^2 = 0^2 = 0$

   So, the inequality becomes: $0 \le |f'(x)| \le 0$ By the Squeeze Theorem, this implies that $|f'(x)|$ must be equal to 0.

4. Deduce the nature of f(x): If $|f'(x)| = 0$, then $f'(x) = 0$ for all $x \in R$. A fundamental result in calculus states that if the derivative of a function is zero everywhere on an interval, then the function must be a constant on that interval. Since $f'(x) = 0$ for all $x \in R$, $f(x)$ must be a constant function. Let $f(x) = C$ for some constant $C$.

5. Use the given condition to find C: We are given that $f(10) = 100$. Since $f(x) = C$, we have $f(10) = C$. Therefore, $C = 100$.

6. Find f(20): Now we know that $f(x) = 100$ for all $x \in R$. To find $f(20)$, we simply substitute $x = 20$ into the function: $f(20) = 100$.

The value of $f(20)$ is 100.