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 derivative equal to zero

To find the turning points, we need to set the derivative ( f'(x) ) equal to zero:

$$ 3.61 e^{2x} \cos(3x + 0.983) = 0 $$

Since ( e^{2x} ) is never zero, we focus on the cosine term:

$$ \cos(3x + 0.983) = 0 $$

2. Solve for ( 3x + 0.983 )

The cosine function equals zero at:

$$ 3x + 0.983 = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z} $$

We can rearrange this to find ( x ):

$$ 3x = \frac{\pi}{2} + k\pi - 0.983 $$

Divide by 3:

$$ x = \frac{1}{3} \left( \frac{\pi}{2} + k\pi - 0.983 \right) $$

3. Calculate the smallest positive ( x )

Start with ( k = 0 ):

$$ x = \frac{1}{3} \left( \frac{\pi}{2} - 0.983 \right) $$

Calculate this value:

• First, compute ( \frac{\pi}{2} \approx 1.5708 ).

Now, substitute and compute:

$$ x = \frac{1}{3} (1.5708 - 0.983) $$

4. Decimal calculation

Calculating it step-by-step:

$$ 1.5708 - 0.983 = 0.5878 $$

Now divide by 3:

$$ x \approx \frac{0.5878}{3} \approx 0.19593 $$

5. Try ( k = 1 ) for the next possibility

$$ x = \frac{1}{3} \left( \frac{\pi}{2} + \pi - 0.983 \right) $$

Calculate this for ( k = 1 ):

$$ x = \frac{1}{3} \left( \frac{3\pi}{2} - 0.983 \right) $$

6. Calculate this value

Using ( \frac{3\pi}{2} \approx 4.7124 ):

$$ x \approx \frac{1}{3} (4.7124 - 0.983) $$

Calculate:

• First, compute ( 4.7124 - 0.983 = 3.7294 ).

Then divide by 3:

$$ x \approx \frac{3.7294}{3} \approx 1.24313 $$

Now we have two potential values for ( x ): ( 0.19593 ) and ( 1.24313 ). The smallest positive value is:

$$ x \approx 0.19593 $$