Question

8. (a) Let $f: (1, +?) \to \mathbb{R}$ be a differentiable function. (i) Assume that $|f'(x)| \le \frac{1}{x^2}$ for all $x > 1$. Show that $\lim_{x \to +\infty} (f(x^5) - f(x^3)) = 0$. [10 marks] (ii) Assume that $|f'(x)| \le \frac{1}{\sqrt{x}}$ for all $x > 1$. Is it necessarily true that $\lim_{x \to +\infty} (f(x^5) - f(x^3)) = 0$? Justify your answer. [3 marks] (b) Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable at some $a \in \mathbb{R}$, satisfying $f(x) \ge f(a)$ for all $x \in (a, a + \frac{1}{2})$. Show that $f'(a) \ge 0$. [4 marks] (c) Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable, such that $f'(0) < 0$ and $f'(1) > 0$. Find the mistake in the following argument: \"The function $f'$ is continuous on $[0, 1]$. Since $f'(0) < 0$ and $f'(1) > 0$, the Intermediate Value Theorem implies that there exists $x_0 \in (0, 1)$ such that $f'(x_0) = 0.$\" [3 marks]

          8. (a) Let $f: (1, +?) \to \mathbb{R}$ be a differentiable function.

(i) Assume that $|f'(x)| \le \frac{1}{x^2}$ for all $x > 1$. Show that
$\lim_{x \to +\infty} (f(x^5) - f(x^3)) = 0$.

[10 marks]

(ii) Assume that $|f'(x)| \le \frac{1}{\sqrt{x}}$ for all $x > 1$. Is it necessarily true that
$\lim_{x \to +\infty} (f(x^5) - f(x^3)) = 0$?
Justify your answer.

[3 marks]

(b) Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable at some $a \in \mathbb{R}$, satisfying
$f(x) \ge f(a)$ for all $x \in (a, a + \frac{1}{2})$.
Show that $f'(a) \ge 0$.

[4 marks]

(c) Let $f: \mathbb{R} \to \mathbb{R}$ be differentiable, such that
$f'(0) < 0$ and $f'(1) > 0$.
Find the mistake in the following argument:
\"The function $f'$ is continuous on $[0, 1]$. Since $f'(0) < 0$ and $f'(1) > 0$, the Intermediate
Value Theorem implies that there exists $x_0 \in (0, 1)$ such that $f'(x_0) = 0.$\"

[3 marks]

8. (a) Let f: (1, +?) →ℝ be a differentiable function.

(i) Assume that |f'(x)| ≤(1)/(x^2) for all x > 1. Show that
limx → +∞ (f(x^5) - f(x^3)) = 0.

[10 marks]

(ii) Assume that |f'(x)| ≤(1)/(√(x)) for all x > 1. Is it necessarily true that
limx → +∞ (f(x^5) - f(x^3)) = 0?
Justify your answer.

[3 marks]

(b) Let f: ℝ→ℝ be differentiable at some a ∈ℝ, satisfying
f(x) ≥ f(a) for all x ∈ (a, a + (1)/(2)).
Show that f'(a) ≥ 0.

[4 marks]

(c) Let f: ℝ→ℝ be differentiable, such that
f'(0) < 0 and f'(1) > 0.
Find the mistake in the following argument:
T̈he function f' is continuous on [0, 1]. Since f'(0) < 0 and f'(1) > 0, the Intermediate
Value Theorem implies that there exists x0 ∈ (0, 1) such that f'(x0) = 0.[3 marks]

Added by Andre M.

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