Let's go through each question step by step.

Problem 4

Find all $x$ values where the function $f(x) = 6x^3 + 5x - 2$ has a tangent line with a gradient equal to 6.

1. To find the values of $x$ where the tangent has a gradient of 6, we need to calculate $f'(x)$ and set it equal to 6.

2. First, find $f'(x)$: $f(x) = 6x^3 + 5x - 2$ Differentiating: $f'(x) = 18x^2 + 5$

3. Set $f'(x) = 6$: $18x^2 + 5 = 6$ $18x^2 = 1$ $x^2 = \frac{1}{18}$ $x = \pm \frac{1}{\sqrt{18}} = \pm \frac{1}{3\sqrt{2}} = \pm \frac{\sqrt{2}}{6}$

Thus, the values of $x$ where the tangent line has a gradient of 6 are: $x = \pm \frac{\sqrt{2}}{6}$

Problem 5

Given $F(x) = f(g(x))$, $g(4) = 6$, $g'(4) = 8$, $f'(4) = 2$, and $f'(6) = 10$, find the value of $F'(4)$.

1. To find $F'(4)$, use the chain rule: $F'(x) = f'(g(x)) \cdot g'(x)$

2. Substitute $x = 4$: $F'(4) = f'(g(4)) \cdot g'(4)$

3. From the information given:

   • $g(4) = 6$
   • $g'(4) = 8$
   • $f'(6) = 10$ (since $g(4) = 6$)

4. Substitute these values: $F'(4) = f'(6) \cdot 8 = 10 \cdot 8 = 80$

So, $F'(4) = 80$.

Problem 6(a)

Find the equation of the tangent line at the given value $f(x) = 3 \cos x + \pi x$ at $x = 0$.

1. First, evaluate $f(0)$: $f(0) = 3 \cos(0) + \pi \cdot 0 = 3$

2. Next, find $f'(x)$: $f(x) = 3 \cos x + \pi x$ Differentiating: $f'(x) = -3 \sin x + \pi$

3. Now, find $f'(0)$: $f'(0) = -3 \sin(0) + \pi = \pi$

4. The equation of the tangent line is: $y = f(0) + f'(0)(x - 0)$ $y = 3 + \pi x$

Thus, the equation of the tangent line at $x = 0$ is: $y = 3 + \pi x$

Would you like further details or explanations on any of these solutions?

Here are some related questions to explore:

1. How would we find the points of inflection for the function $f(x) = 6x^3 + 5x - 2$?
2. What is the process for finding the second derivative of a composite function like $F(x) = f(g(x))$?
3. How would the tangent line equation change if we used a different point for evaluation?
4. Can you find other values of $x$ for which $f(x) = 3 \cos x + \pi x$ has a horizontal tangent?
5. What would be the derivative of $f(x) = 3 \cos x + \pi x$ if it included a higher degree term?

Tip: When dealing with composite functions, always apply the chain rule carefully, evaluating inner and outer derivatives accurately to avoid errors.