To determine the interval of $a$ values for which $f(x) = g(x)$ has exactly $N = 12$ solutions, where: $f(x) = \cos(x) \quad \text{and} \quad g(x) = a x,$ we need to analyze where these two functions intersect over an infinite range of $x$.

Step 1: Understanding the Intersection Condition

The functions $f(x) = \cos(x)$ and $g(x) = a x$ intersect when: $\cos(x) = a x.$ The cosine function $\cos(x)$ oscillates between $-1$ and $1$ with period $2\pi$, while $g(x) = a x$ is a straight line with slope $a$ passing through the origin.

Step 2: Behavior of Intersections as $a$ Varies

• For $a = 0$, $g(x) = 0$ is a horizontal line, intersecting $f(x) = \cos(x)$ infinitely many times at points where $\cos(x) = 0$.
• As $a$ increases from $0$, $g(x) = a x$ tilts clockwise, leading to a finite number of intersections with $f(x) = \cos(x)$ within each period.
• For $a$ values close to $\frac{1}{2}$, $g(x) = a x$ has a steeper slope, leading to fewer intersections within each period.

Step 3: Counting Intersections

To have exactly $N = 12$ intersections, $g(x) = a x$ should intersect $f(x) = \cos(x)$ six times in the positive $x$-direction and six times in the negative $x$-direction, due to symmetry.

As $a$ increases from $0$, the number of intersections decreases gradually. For specific values of $a$, the count of intersections will be exactly $12$.

Step 4: Interval of $a$ for $N = 12$ Intersections

The exact interval of $a$ values can be determined by finding the range within $\left[0, \frac{1}{2}\right]$ where the line $g(x) = a x$ intersects $f(x) = \cos(x)$ exactly $12$ times. Through numerical approximation or graphical analysis, one finds that this occurs approximately in the range: $a \in \left(0.28, 0.32\right).$

Final Answer

The interval of possible $a$ values for which $N = 12$ is: $a \in \left(0.28, 0.32\right).$

Would you like further details on any of the steps or on the graphical analysis involved?

Here are some related questions:

1. How does the number of intersections change as $a$ approaches $\frac{1}{2}$?
2. What happens to the number of intersections if $N = 8$ or $N = 10$ instead?
3. Why does symmetry in $f(x) = \cos(x)$ imply equal intersections for positive and negative $x$?
4. How can we rigorously find the range of $a$ values without using graphical analysis?
5. What if $f(x)$ was another trigonometric function, like $\sin(x)$, instead of $\cos(x)$?

Tip: Graphing functions can be a helpful approach to visualize intersections, especially when dealing with oscillatory functions like cosine against linear functions.