Given: $f(x) = x^3 + 4x^2 - 7x - 10$ 8.1 Write down the y-intercept of $f$ - NSC Mathematics - Question 8 - 2023 - Paper 1

To show that $2$ is a root, we will substitute $x = 2$ into $f(x)$:

$f(2) = 2^3 + 4(2)^2 - 7(2) - 10 = 8 + 16 - 14 - 10 = 0$

Since $f(2) = 0$, $2$ is indeed a root of the equation.

Since $2$ is a root, we can factor $f(x)$ using synthetic division:

1. Perform synthetic division of $f(x)$ by $(x - 2)$:

$f(x) = (x - 2)(x^2 + 6x + 5)$

2. The quadratic $x^2 + 6x + 5$ can be further factored as:

$(x + 1)(x + 5)$

3. Therefore, the complete factorisation is:

$f(x) = (x - 2)(x + 1)(x + 5)$

To draw the graph:

1. Plot the y-intercept at $(0, -10)$.
2. Identify the x-intercepts by solving $f(x) = 0$, which occur at $x = -5, -1, 2$.
3. Mark the turning points at $(0.7; 12.6)$ and $(-3.4; 20.8)$.
4. Sketch the curve that connects these points, being mindful of the behavior of cubic functions around turning points.

To determine where $f'(x) < 0$, analyze the graph of $f$. It shows that the function is decreasing between the turning points, indicating where the derivative is negative.

The gradient of the tangent is minimum at the point where the derivative is zero. This occurs at the turning points of the graph, which can be identified at $x = -3.4$ and $x = 0.7$.

To satisfy f'(x) ullet f''(x) ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ }:

1. Identify intervals where the product of the derivative and second derivative is less than or equal to zero, typically at critical points and boundaries of increasing/decreasing intervals.