Let's analyze the problem step by step. The function given is:

$f(x) = -x^3 + 6x^2 - 9x + 1$

i. Expression for $f'(x)$

To find the derivative of $f(x)$, apply standard differentiation rules to each term:

$f'(x) = \frac{d}{dx} (-x^3 + 6x^2 - 9x + 1)$

This yields:

$f'(x) = -3x^2 + 12x - 9$

ii. Turning points

To find the turning points, set $f'(x) = 0$ and solve for $x$:

$-3x^2 + 12x - 9 = 0$

Divide through by $-3$:

$x^2 - 4x + 3 = 0$

Factorize:

$(x - 3)(x - 1) = 0$

Thus, $x = 3$ and $x = 1$ are the critical points. To confirm whether these are maxima or minima, we will check the second derivative in the next step.

iii. Sketch the graph of the function

To sketch the graph, we need the behavior of the function at the critical points, intercepts, and end behavior. Here's a summary:

1. Critical points: We found $x = 1$ and $x = 3$.
2. Y-intercept: $f(0) = 1$ (the constant term in the polynomial).
3. Behavior as $x \to \infty$: Since the leading term is $-x^3$, the function tends towards $-\infty$ as $x \to \infty$ and $\infty$ as $x \to -\infty$.

A rough sketch would show a cubic function with a local maximum and minimum between $x = 1$ and $x = 3$.

iv. Intervals where the function is increasing and decreasing

To determine where the function is increasing or decreasing, analyze the sign of $f'(x) = -3x^2 + 12x - 9$.

• Factorized form: $f'(x) = -3(x - 3)(x - 1)$.

• For $x < 1$: Both factors are negative, so $f'(x) > 0$ (function is increasing).

• For $1 < x < 3$: One factor is negative, one is positive, so $f'(x) < 0$ (function is decreasing).

• For $x > 3$: Both factors are positive, so $f'(x) > 0$ (function is increasing).

v. Find the relative extrema for the function $f(x)$

Evaluate $f(x)$ at the critical points $x = 1$ and $x = 3$:

$f(1) = -(1)^3 + 6(1)^2 - 9(1) + 1 = -1 + 6 - 9 + 1 = -3$ $f(3) = -(3)^3 + 6(3)^2 - 9(3) + 1 = -27 + 54 - 27 + 1 = 1$

• At $x = 1$, $f(x) = -3$ (local minimum).
• At $x = 3$, $f(x) = 1$ (local maximum).

Summary:

1. $f'(x) = -3x^2 + 12x - 9$
2. Turning points are at $x = 1$ and $x = 3$.
3. The function increases for $x < 1$ and $x > 3$, and decreases between $1 < x < 3$.
4. Relative minimum at $(1, -3)$ and relative maximum at $(3, 1)$.

Would you like further details or have any questions on this?

Here are 5 related questions for further practice:

1. How does the second derivative test confirm the nature of turning points?
2. What are the concavity and points of inflection for this function?
3. How would the graph change if the coefficient of $x^3$ were positive?
4. Can we find any real roots for $f(x) = 0$? If yes, how?
5. How does the behavior of a cubic function at infinity differ from that of a quadratic?

Tip: When sketching a function, always check both critical points and inflection points to get a better understanding of its overall shape!