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Verify Cauchy Mean Value Theorem for f(x)= x3 , g(x)= x2 in the interval \[-1,1\]. Find the value of c. (Add graphical explanations)   Solve by Lagrange  multiplier method:  Maximize f(x,y) =x2 + y2 with constraint: (x-1)2 +4y2 =4.   Solve by Lagrange  multiplier method:  Maximize f(x,y) =x2 + y2 with constraint: (x-1)2 +4y2 =4. Find radius of convergence of the curve : x=a cos\^3⁡〖θ,〗  y=a sin\^3⁡θ.      at     θ=π/4.   Consider the following system of equations: x-3y+3z = -1, x-3y+4z= 1, -2x+4y-6z= k. Find different values of k, such that the system has (i) no solution (ii) infinite solution (iii) unique solution.   Let the eigenvalues and eigenvectors of a 2\*2 matrix A are 1,-2 and x1 & x2 respectively. Then find the eigenpairs for the matrix,  B=A2-3A+4I.   ∫\_0\^(π/2)▒〖√(sin⁡θ ).cos\^5⁡θ  〗 dθ=?