Hence, or otherwise, find the smallest positive value of x, where x is in radians, for which the curve with equation y = f(x) has a turning point.

Steps to Solve

1. Set the equation for turning points

Turning points occur where the derivative $f'(x)$ equals zero or is undefined. We need to find the solutions to:

$$ 2 \cos 3x - 3 \sin 3x = 3.61 \cos(3x + 0.983) $$

2. Rearrange the equation

We can bring everything to one side:

$$ 2 \cos 3x - 3 \sin 3x - 3.61 \cos(3x + 0.983) = 0 $$

3. Use a substitution method

To simplify our calculations, let's express $\cos(3x + 0.983)$ in terms of $\cos(3x)$ and $\sin(3x)$ using the angle addition formula:

$$ \cos(3x + 0.983) = \cos 3x \cos 0.983 - \sin 3x \sin 0.983 $$

4. Plug in the values

Substituting back into the equation gives us:

$$ 2 \cos 3x - 3 \sin 3x - 3.61 (\cos 3x \cos 0.983 - \sin 3x \sin 0.983) = 0 $$

5. Combine like terms

This equation can be further simplified by collecting like terms:

$$ (2 - 3.61 \cos 0.983) \cos 3x + (-3 + 3.61 \sin 0.983) \sin 3x = 0 $$

6. Create a system of equations

For a non-trivial solution with $A \cos 3x + B \sin 3x = 0$, we can use the condition:

$$ A = 2 - 3.61 \cos 0.983 = 0 $$

$$ B = -3 + 3.61 \sin 0.983 = 0 $$

7. Solve for 3x

Find $3x$ from $A$ and $B$. Rearranging gives:

$$ \tan(3x) = \frac{A}{B} = \frac{2 - 3.61 \cos 0.983}{-3 + 3.61 \sin 0.983} $$

8. Determine the value of x

To find $x$, we solve for $3x$ first, and then divide by 3. The smallest positive solution for $3x$ will be:

$$ 3x = \arctan\left(\frac{A}{B}\right) $$

9. Get x in radians

Finally, convert to $x$:

$$ x = \frac{\arctan\left(\frac{A}{B}\right)}{3} $$