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Problem: If there exists a line not parallel to the coordinate axes or $y=\pm x$ that does not intersect the curve
$$2\cos x - \cos y = a (a \ge 0)$$

Find the range of $a$.

This is a problem I came up with myself. I've tried it on Geogebra, which indicates that $\frac{3}{2} \lt a \le 3$. However, I haven't figured out how to solve it rigorously. Here is a figure to show the case $a=1.5$, and the line is $y=2x$, which seems tangent to the curve.

I hope to solve this problem using high school methods, including basic derivatives and some analytic geometry techniques.

Update: Question remaining to be answered: $$f(x) = 2\cos x - \cos (\frac{1}{2}x+b)$$ $$f(x) = 2\cos x - \cos (\frac{3}{2}x+b)$$ $$f(x) = 2\cos x - \cos (\frac{5}{2}x+b)$$

Are the maximums of these three functions larger than $1.5$ for any $b$?

Problem: If there exists a line not parallel to the coordinate axes or $y=\pm x$ that does not intersect the curve
$$2\cos x - \cos y = a (a \ge 0)$$

Find the range of $a$.

This is a problem I came up with myself. I've tried it on Geogebra, which indicates that $\frac{3}{2} \lt a \le 3$. However, I haven't figured out how to solve it rigorously. Here is a figure to show the case $a=1.5$, and the line is $y=2x$, which seems tangent to the curve.

I hope to solve this problem using high school methods, including basic derivatives and some analytic geometry techniques.

Problem: If there exists a line not parallel to the coordinate axes or $y=\pm x$ that does not intersect the curve
$$2\cos x - \cos y = a (a \ge 0)$$

Find the range of $a$.

This is a problem I came up with myself. I've tried it on Geogebra, which indicates that $\frac{3}{2} \lt a \le 3$. However, I haven't figured out how to solve it rigorously. Here is a figure to show the case $a=1.5$, and the line is $y=2x$, which seems tangent to the curve.

I hope to solve this problem using high school methods, including basic derivatives and some analytic geometry techniques.

Update: Question remaining to be answered: $$f(x) = 2\cos x - \cos (\frac{1}{2}x+b)$$ $$f(x) = 2\cos x - \cos (\frac{3}{2}x+b)$$ $$f(x) = 2\cos x - \cos (\frac{5}{2}x+b)$$

Are the maximums of these three functions larger than $1.5$ for any $\cos b>0.5$?

Problem: If there exists a line not parallel to the coordinate axes or $y=\pm x$ that does not intersect the curve
$$2\cos x - \cos y = a (a \ge 0)$$

Find the range of $a$.

This is a problem I came up with myself. I've tried it on Geogebra, which indicates that $\frac{3}{2} \lt a \le 3$. However, I haven't figured out how to solve it rigorously. Here is a figure to show the case $a=1.5$, and the line is $y=2x$, which seems tangent to the curve.

I hope to solve this problem using high school methods, including basic derivatives and some analytic geometry techniques.

Update: Question remaining to be answered: $$f(x) = 2\cos x - \cos (\frac{1}{2}x+b)$$ $$f(x) = 2\cos x - \cos (\frac{3}{2}x+b)$$ $$f(x) = 2\cos x - \cos (\frac{5}{2}x+b)$$

Are the maximums of these three functions larger than $1.5$ for any $b$?

Problem: If there exists a line not parallel to the coordinate axes or $y=\pm x$ that does not intersect the curve
$$2\cos x - \cos y = a (a \ge 0)$$

Find the range of $a$.

This is a problem I came up with myself. I've tried it on Geogebra, which indicates that $\frac{3}{2} \lt a \le 3$. However, I haven't figured out how to solve it rigorously. Here is a figure to show the case $a=1.5$, and the line is $y=2x$, which seems tangent to the curve.

I hope to solve this problem using high school methods, including basic derivatives and some analytic geometry techniques.

Update: Question remaining to be answered: $$f(x) = 2\cos x - \cos (\frac{1}{2}+b)$$ $$f(x) = 2\cos x - \cos (\frac{3}{2}+b)$$ $$f(x) = 2\cos x - \cos (\frac{5}{2}+b)$$

Are the maximums of these three function larger than $1.5$ for any $\cos b>0.5$?

Problem: If there exists a line not parallel to the coordinate axes or $y=\pm x$ that does not intersect the curve
$$2\cos x - \cos y = a (a \ge 0)$$

Find the range of $a$.

This is a problem I came up with myself. I've tried it on Geogebra, which indicates that $\frac{3}{2} \lt a \le 3$. However, I haven't figured out how to solve it rigorously. Here is a figure to show the case $a=1.5$, and the line is $y=2x$, which seems tangent to the curve.

I hope to solve this problem using high school methods, including basic derivatives and some analytic geometry techniques.

Update: Question remaining to be answered: $$f(x) = 2\cos x - \cos (\frac{1}{2}x+b)$$ $$f(x) = 2\cos x - \cos (\frac{3}{2}x+b)$$ $$f(x) = 2\cos x - \cos (\frac{5}{2}x+b)$$

Are the maximums of these three functions larger than $1.5$ for any $\cos b>0.5$?