Can the graphs of a polynomial of degree 20 and the function y=1/x⁴⁰ have exactly 30 points of intersection?

Great breakdown! To add a bit more detail:

We're looking for 30 points of intersection between a degree-20 polynomial $P(x)$ and the rational function <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>40</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">y = \frac{1}{x^{40}}</annotation></semantics></math>y=x401​. Setting them equal gives:

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>40</mn></msup></mfrac><mo>⇒</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>x</mi><mn>40</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P(x) = \frac{1}{x^{40}} \Rightarrow P(x)x^{40} = 1</annotation></semantics></math>P(x)=x401​⇒P(x)x40=1

So, we get the equation:

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>x</mi><mn>40</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">P(x)x^{40} - 1 = 0</annotation></semantics></math>P(x)x40−1=0

This is a degree-60 equation (since <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>deg</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>x</mi><mn>40</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mn>20</mn><mo>+</mo><mn>40</mn><mo>=</mo><mn>60</mn></mrow><annotation encoding="application/x-tex">\deg(P(x)x^{40}) = 20 + 40 = 60</annotation></semantics></math>deg(P(x)x40)=20+40=60). So in theory, up to 60 real or complex roots (intersections), but we’re only interested in whether exactly 30 real intersections are possible.

To analyze that, we look at the number of local extrema needed to support 30 real roots. As pointed out, 30 roots imply at least 29 local extrema (by Rolle’s Theorem). So we check how many critical points the function <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>x</mi><mn>40</mn></msup><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f(x) = P(x)x^{40} - 1</annotation></semantics></math>f(x)=P(x)x40−1 can have.

The derivative is:

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>40</mn><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>x</mi><mn>39</mn></msup><mo>+</mo><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>x</mi><mn>40</mn></msup><mo>=</mo><msup><mi>x</mi><mn>39</mn></msup><mo stretchy="false">(</mo><mn>40</mn><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>x</mi><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x) = 40P(x)x^{39} + P'(x)x^{40} = x^{39}(40P(x) + xP'(x))</annotation></semantics></math>f′(x)=40P(x)x39+P′(x)x40=x39(40P(x)+xP′(x))

• <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>39</mn></msup></mrow><annotation encoding="application/x-tex">x^{39}</annotation></semantics></math>x39 gives one root at $x=0$

• <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>40</mn><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>x</mi><msup><mi>P</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">40P(x) + xP'(x)</annotation></semantics></math>40P(x)+xP′(x) is a degree-20 polynomial (since $P(x)$ is degree 20), so at most 20 roots.

That means the derivative has at most 21 real roots → at most 21 extrema, which is less than the 29 needed.

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Posted by lyraellington12 on 2025-05-27 04:54:44