Solve for x cos(x)=x^4-x

Problem

cos(x)=x4−x

Solution

1. Identify the type of equation. This is a transcendental equation involving a trigonometric function and a polynomial, which cannot be solved for x using standard algebraic methods.

2. Rearrange the equation into a root-finding form by moving all terms to one side.

cos(x)−x4+x=0

3. Define a function ƒ(x) to analyze the behavior and find where it equals zero.

ƒ(x)=cos(x)−x4+x

4. Analyze the function at specific points to find intervals where a root exists using the Intermediate Value Theorem.

ƒ(0)=cos(0)−0+0=1

ƒ(1)=cos(1)−1+1=cos(1)≈0.5403

ƒ(2)=cos(2)−2+2=cos(2)−14≈−14.4161

5. Observe that since ƒ(1)>0 and ƒ(2)<0 there must be a root in the interval (1,2)

6. Apply a numerical method such as Newton's Method, (x_n+1)=(x_n)−ƒ((x_n))/(ƒ((x_n))′) where ƒ(x)′=−sin(x)−4*x3+1

7. Iterate starting with an initial guess (x_0)=1.1

(x_1)=1.1−(cos(1.1)−1.1+1.1)/(−sin(1.1)−4*(1.1)3+1)≈1.1164

(x_2)=1.1164−ƒ(1.1164)/(ƒ(1.1164)′)≈1.1163

8. Check for other roots. For large negative x x4 dominates and makes the expression negative.

ƒ*(−1)=cos(−1)−(−1)4+(−1)=cos(1)−1−1≈−1.4597

Since ƒ(0)=1 and ƒ*(−1)<0 there is a root in the interval (−1,0)

9. Iterate for the second root starting with (x_0)=−0.5

(x_1)=−0.5−(cos(−0.5)−(−0.5)4+(−0.5))/(−sin(−0.5)−4*(−0.5)3+1)≈−0.6865

(x_2)≈−0.7047

(x_3)≈−0.7050

Final Answer

cos(x)=x4−x⇒x≈−0.7050,1.1163

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