First, the user is asking about the right plot in the image. I need to find how many local maxima there are along the tau axis when K=1.5. The final answer should be an exact integer.

Looking at the image, there are two 3D plots. The right one is the one I need to focus on. The axes are labeled: K, tau, and the z-axis seems to be some value that the surface represents, ranging from 0 to 2.

A local maximum is a point where the value is higher than all nearby points. Along the tau axis, I need to consider slices where K is fixed at 1.5.

So, I should find the value of the function at K=1.5 for different tau values and identify where it has local maxima.

From the plot, when K=1.5, I need to look at the curve along tau. Since it's a 3D surface, fixing K=1.5 means I'm looking at a cross-section parallel to the tau-z plane.

Let me describe what I see in the right plot. The surface has peaks and valleys. The color gradient indicates height: red for higher values, blue for lower.

Now, for K=1.5, I need to trace the surface at that K value. Looking at the grid, K ranges from 0 to 1.5 on the x-axis, so K=1.5 is at the far right.

In the plot, the K axis goes from 0 to 1.5, so K=1.5 is the maximum K value.

I should find where the surface is at K=1.5 and see how it varies with tau.

tau ranges from 0 to 1 on the y-axis.

So, at K=1.5, tau from 0 to 1, what does the surface look like?

From the image, at K=1.5, the surface has some points. I need to see the z-values as a function of tau for fixed K=1.5.

Since it's a surface plot, I can imagine slicing it at K=1.5.

Looking at the right plot, when K=1.5, which is the rightmost part of the K axis, the surface seems to have a certain shape.

I think I need to look at the values where the surface is evaluated. There are black dots on the surface; those might be data points.

The plot has a mesh surface and black dots that might represent sampled points or something.

But for finding local maxima, I should consider the surface itself.

Perhaps the black dots are the actual data points, and the surface is an interpolation.

The question is about local maxima, so I need to find peaks in the surface along tau at K=1.5.

Let me try to sketch what I see.

In the right plot, for K from 0 to 1.5, and tau from 0 to 1, z from 0 to 2.

At K=1.5, which is the end, the surface seems to have a peak or something.

I should look at the tau axis variation.

Since K is fixed at 1.5, I need the function z(tau) at K=1.5.

From the plot, at K=1.5, as tau varies, z varies.

Let me see the range.

For example, when tau=0, at K=1.5, what is z? From the plot, at (K,tau) = (1.5,0), it seems z is around 1.0 or something, but I need to be precise.

This is visual, so I have to estimate.

Perhaps the black dots can help me identify points.

There are several black dots. At K=1.5, I see a few dots.

Let me list the points at K=1.5.

K=1.5 is the last value on the x-axis.

In the right plot, at K=1.5, I see multiple black dots at different tau values.

For instance, there is a dot at tau around 0.25, z around 1.8; another at tau=0.5, z=1.6; tau=0.75, z=1.4; and tau=1.0, z=1.2 or something. I need to estimate the coordinates.

This is tricky. Let's look at the surface grid.

The surface is smooth, so I can see the height.

At K=1.5, the surface starts at tau=0, z≈1.0, then as tau increases, it might go up or down.

From the plot, at K=1.5, when tau=0, z is about 1.0. Then as tau increases, it seems to increase to a peak around tau=0.25, z≈1.8, then decrease to tau=0.5, z≈1.6, then decrease further to tau=0.75, z≈1.4, and to tau=1.0, z≈1.2.

But is that a local maximum? At tau=0.25, z=1.8, which is higher than neighbors.

Then at tau=0, z=1.0, which is lower, and at tau=0.5, z=1.6 < 1.8, so yes, a local max at tau≈0.25.

Now, are there other local maxima? From tau=0.5 to tau=1.0, it's decreasing: z=1.6 at tau=0.5, 1.4 at tau=0.75, 1.2 at tau=1.0, so no other peaks.

But at tau=0, is it a local max? At tau=0, z=1.0, and for tau>0, z increases, so it's not a local max; it's a local min or something.

Similarly, at tau=1.0, z=1.2, and for tau<1.0, z is higher, so not a local max.

So only one local maximum at around tau=0.25.

But let's confirm with the black dots. The black dots might be the data points, so I should see where they are.

In the plot, at K=1.5, there are black dots at various points.

For example, one dot at (K,tau) = (1.5, 0.25) with high z, say 1.8; another at (1.5, 0.5) with z=1.6; (1.5, 0.75) with z=1.4; and (1.5, 0) with z=1.0 or so.

Also, there might be more, but from the surface, it seems continuous.

Perhaps I should consider if there are multiple peaks.

Another way: a local maximum along the tau axis for fixed K means that for that tau, z(tau) is higher than for neighboring tau values at the same K.

So at K=1.5, I need to see the function z(tau).

From the surface, it increases to a peak at tau≈0.25 and then decreases, so only one local maximum.

But let's see the left plot for comparison. In the left plot, at K=1.5, it might be different, but the question is about the right plot.

The user said "for the right plot", so I focus on that.

In the right plot, at K=1.5, the surface has a single hump, so one local max.

But is there a local max at the endpoints? Usually, for endpoints, we don't consider them as local maxima unless specified, because they are boundaries.

For example, at tau=0, since tau=0 is the boundary, and z increases for tau>0, it's not a local maximum. Similarly for tau=1.

So only interior points.

Thus, at K=1.5, there is one local maximum along tau.

But let's make sure about the values.

Perhaps I can count the number of black dots at high z.

At K=1.5, the highest dot is at tau=0.25, z=1.8, which is a peak.

Then lower at other tau.

So I think it's one.

The question is "how many local maxima can be observed", and it should be an integer.

But in my reasoning, it's one.

Let me double-check the plot.

Looking at the image, in the right plot, for K=1.5, the surface is shown, and it has one clear peak when tau is around 0.25.

For example, at tau=0.25, z is maximum, and it decreases on both sides in tau.

Since K is fixed, we vary tau.

Yes.

Perhaps at other K, there are multiple, but at K=1.5, it's single.

Another thing: the surface might have multiple peaks, but at K=1.5, it seems monotonic in the sense of one peak.

Let's see the tau axis: from 0 to 1.

At K=1.5, tau=0: z≈1.0

tau=0.25: z≈1.8

tau=0.5: z≈1.6

tau=0.75: z≈1.4

tau=1.0: z≈1.2

So the values: 1.0, 1.8, 1.6, 1.4, 1.2

Clearly, at tau=0.25, it's a peak since 1.8 > 1.0 and 1.8 > 1.6.

No other points are higher than neighbors.

For example, tau=0.5: 1.6, but 1.6 < 1.8 and 1.6 > 1.4? 1.6 > 1.4, but since 1.6 < 1.8, it's not a local max; it's decreasing from the peak.

Similarly, others are not local maxima.

So only one.

But the user said "along the tau axis when K=1.5", and I think I have it.

The question is: "how many local maxima can be observed along the tau axis when K=1.5 for the right plot?"

And I think it's one.

But let's see if there is any other point.

Perhaps at the boundaries, but no.

Another thought: is K=1.5 exactly the edge? But since it's a point, we can evaluate.

In the plot, K=1.5 is included, and tau from 0 to 1.

Perhaps I should consider the derivative or something, but visually, it's clear.

Maybe the black dots indicate specific points, and I need to see if among them there are local maxima.

But the surface is smooth, so it shouldn't matter.

For example, at K=1.5, there is a black dot at tau=0.25, high z, and others lower, so it's a local max.

No other dots at high z at other tau.

So I think it's one.

But the answer should be an integer, and I think it's 1.

But let's confirm the left plot quickly to see if it's different.

In the left plot, at K=1.5, the surface is different. It seems lower and flatter or something.

At K=1.5 in left plot, z is around 1.0 or less, and it might not have a clear peak.

But for the right plot, it's higher.

The user specified the right plot.

Perhaps I misread the axes.

The axes: K is x-axis, tau is y-axis, z is color and height.

In the image, for right plot, at K=1.5, which is x=1.5, y from 0 to 1.

The surface at (1.5, y) for y in [0,1].

As y increases, z first increases to about y=0.25, then decreases.

So one local maximum.

Perhaps at y=0, but z is lower.

So I think it's one.

But the question says "along the tau axis", which is the y-axis, so varying tau.

Yes.

So the number is 1.

But the user might expect a different answer, or I need to be careful.

Another idea: "local maxima" might refer to points where the surface has a peak, but in 3D, a local maximum would be a point higher than all adjacent points, but here it's "along the tau axis", which I think means for fixed K, varying tau.

The phrase: "how many local maxima can be observed along the tau axis when K=1.5"

I think it means: for fixed K=1.5, how many local maxima in the direction along the tau axis, i.e., in the tau direction.

In other words, for the slice at K=1.5, how many local max in tau.

Yes.

Sometimes "along" might mean something else, but I think this is standard.

Perhaps it means finding points where the partial derivative with respect to tau is zero, but visually, from the plot, it's clear.

So I think the answer is 1.

But let's see the final instruction: "Your final answer must be an exact integer."

And in the context, it should be 1.

But perhaps I should write it as 1.

But let's see if there is any other local max.

For example, at tau=0.5, z=1.6, but it's less than at tau=0.25, and greater than at tau=0.75, but since it's not greater than left neighbor, it's not a local maximum; it's a local minimum or saddle if we consider 3D, but for fixed K, it's not a local max in tau.

In one dimension, for a function f(tau), a local maximum at tau=c if f(c) >= f(c') for all c' near c.

At tau=0.5, f(0.5) =1.6, f(0.25)=1.8 >1.6, so not local max.

Similarly, only at tau=0.25 it is higher.

So I think it's one.

Perhaps the gate G or something, but no, this is a different question.

The image is about two surfaces, probably from some optimization or function.

But for this, I think answer is 1.

So I'll go with that.