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### Part a: Minimum Turning Point

The function is given as \( f(x) = \sqrt{x^2 + 24} \).

To find the minimum turning point, we need to first differentiate the function with respect to \( x \), and then solve for when the derivative equals zero to locate the turning point.

1. \( f'(x) \) is the derivative of \( f(x) = \sqrt{x^2 + 24} \).

2. After finding \( f'(x) \), set it equal to 0 and solve for \( x \). This will give us the \( x \)-coordinate of the turning point.

3. Plug that value back into the original function to find the \( y \)-coordinate.

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### Part b: Oblique Asymptotes

To show that \( y = \pm x \) are the oblique asymptotes:

1. Examine the behavior of \( f(x) = \sqrt{x^2 + 24} \) as \( x \to \infty \) and \( x \to -\infty \).
   
2. Simplify the function for large values of \( x \) by factoring out \( x^2 \) inside the square root, and demonstrate that the function approaches \( y = |x| \) for large values of \( x \), which confirms the oblique asymptotes.

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### Part c: Linear Equations of Tangents and Segments

#### Part (i): Tangent BC

From the figure, the line BC is tangent to the curve \( y = \sqrt{x^2 + 24} \) at \( x = 3 \).

1. First, find the derivative \( f'(x) \) to get the slope of the tangent at \( x = 3 \).

2. Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point on the curve, we can find the equation of the tangent line.

#### Part (ii): Line Segment EB

For the line segment EB, given that it is perpendicular to BC and passes through points E and B:

1. Use the fact that perpendicular lines have slopes that are negative reciprocals to find the slope of EB.

2. Again, apply the point-slope form to determine the equation of EB.

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Here are 5 related questions to consider:

1. How does the square root function influence the behavior of the graph?
2. What role do derivatives play in identifying turning points?
3. How do asymptotes describe the behavior of a function at extreme values?
4. How can we verify the perpendicularity of two lines based on their slopes?
5. How would the shape of the graph change if the constant inside the square root was different?

### Tip:
When solving for turning points using derivatives, always check if the second derivative can confirm whether it's a minimum or maximum.

A particle is moving along a trajectory with the function f(x) = √(x² + 24). Part a: Find the coordinates of the minimum turning point. Part b: Show that the lines y = ± x are the linear oblique asymptotes for the graph of the function. Part c: Determine (i) the linear equation of the tangent BC and (ii) the linear equation of the linear segment EB.

Solve for the minimum turning point of f(x) = √(x² + 24) using differentiation and show that y = ±x are the oblique asymptotes. Also, determine the linear equations for tangent BC and segment EB from the graph.

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