For each of the following functionals, find the Gâteaux differential and use it to derive the associated Euler-Lagrange equation. In each case, be sure to specify all boundary conditions that the stationary path must satisfy. •1 1 S[y] = v(0) + ½ √² dx (2y² + x²y²), y(1) = 2.

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Q: Why'd you choose 1/5?

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A: According to the given data, find the solution:

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A: We have

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Q: do the second question

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A: find where the function is increasing

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Q: 1. Show that 2! 4! 6! (2n)! ≥ ((n + 1)!)" for ne z+. . ...

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A: We have to minimizesubject toAssumeLets use method of Lagrange multipliers.

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A: denote the region trapped between the cone  and the plane We have to compute

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Q: 4.8.) Find the SVD of the matrix A= 322 23-2

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Q: 7. Find the inverse Laplace transform for the F(s): (s - 2)e-2s s²4s +3 Check the denominator also.…

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Q: Use Fubini's Theorem to evaluate 14x²x 00 ¡¡₁. 00 4xexy dx dy. 4xexy dx dy = (Type an exact answer.)

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Q: 7. L { cos² t}

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Q: 3. Suppose p = 2 (mod 5) and p is a prime number. Show that if x5 = 1 (mod p), then x = 1 (mod p).

A: We shall use basic modular arithmetic for solving this problem

Q: Let x₁ = √P (P >0) and XN+1=√P+X (N21). Show that (XN) converges and find the limit.

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