Here are the answers to Exercise 5.1 based on standard calculus rules and the given function:

1. General conditions

(a) A function $f$ is increasing on $(a, b)$ if $f'(x) > 0$ whenever $a < x_1 < x_2 < b$.

(b) A function $f$ is decreasing on $(a, b)$ if $f'(x) < 0$ whenever $a < x_1 < x_2 < b$.

(c) A function $f$ is concave up on $(a, b)$ if $f'$ is increasing on $(a, b)$ or equivalently if $f''(x) > 0$ on $(a, b)$.

(d) If $f''(a)$ exists and $f$ has an inflection point at $x = a$, then $f''(a) = 0$.

2. Function Analysis

Given $f(x) = 0.1(x^3 - 3x^2 - 9x)$, we know:

• $f'(x) = 0.3(x + 1)(x - 3)$
• $f''(x) = 0.6(x - 1)$

(a) Solutions to $f'(x) = 0$:
From $f'(x) = 0.3(x + 1)(x - 3) = 0$,
$x = -1$ and $x = 3$.

(b) Intervals where $f$ is increasing:
To determine increasing/decreasing intervals:

• $f'(x) > 0$ when $(x + 1)(x - 3) > 0$.

Sign analysis:

• $x < -1$: Both factors negative → $f'(x) > 0$
• $-1 < x < 3$: Opposite signs → $f'(x) < 0$
• $x > 3$: Both factors positive → $f'(x) > 0$.

Thus, $f$ is increasing on $(-\infty, -1) \cup (3, \infty)$.

(c) Intervals where $f$ is concave down:
From $f''(x) = 0.6(x - 1)$:

• $f''(x) > 0$ when $x > 1$ → concave up.
• $f''(x) < 0$ when $x < 1$ → concave down.

Thus, $f$ is concave down on $(-\infty, 1)$.

(d) Inflection point:
An inflection point occurs when $f''(x) = 0$.
From $f''(x) = 0.6(x - 1)$:

• $x = 1$.

Thus, $x = 1$ is the inflection point on the graph of $f$.

Final Answers Summary:

1. General conditions:

   • (a) $f'(x) > 0$
   • (b) $f'(x) < 0$
   • (c) $f''(x) > 0$
   • (d) $f''(a) = 0$

2. Function Analysis:

   • (a) $x = -1$ and $x = 3$
   • (b) $(-\infty, -1) \cup (3, \infty)$
   • (c) $(-\infty, 1)$
   • (d) $x = 1$

Related Questions:

1. How do you determine the concavity of a function?
2. What is the significance of the first derivative test for increasing/decreasing intervals?
3. How can inflection points be identified from the second derivative?
4. What is the relationship between critical points and the derivative of a function?
5. How do you analyze intervals for sign changes in the derivative?

Tip: Always perform a sign test on the derivative $f'(x)$ to confirm increasing and decreasing behavior, and similarly for $f''(x)$ to analyze concavity.