math

1. Consider the function f(x) = x^3 + 2x^2 - 3x + 1. Find the local maximum and minimum values of f(x).
2. Prove or disprove: The sum of two irrational numbers is always irrational.
3. Find the limit of (1 + 1/n)^n as n approaches infinity.
4. Prove the Maclaurin series expansion for sin(x) using mathematical induction.
5. Determine the radius of convergence for the power series representation of the function f(x) = ln(1 + x).
6. Solve the differential equation dy/dx = y^2 - x^2, where y(0) = 1.
7. Prove the convergence of the sequence defined by a_n = (n^2)/(2^n).
8. Determine the area enclosed by the curve y = x^3 - 4x^2 + 6x and the x-axis.
9. Find the eigenvalues and corresponding eigenvectors of the matrix A = [2 1; 4 3].
10. Prove the Cauchy-Schwarz inequality for vectors in an inner product space.
11. Solve the recurrence relation a_n = 3a_(n-1) - 2a_(n-2), where a_0 = 1 and a_1 = 3.
12. Compute the double integral ∬(x^2 + y), where the region R is bounded by y = 2x, y = 4 - x, and x = 0.
13. Show that the divergence of the curl of a vector field is always zero.
14. Find the Fourier series representation of the function f(x) = x^2 on the interval [-π, π].
15. Prove that the sum of an infinite geometric series with first term a and common ratio r, where |r| < 1, is given by S = a/(1 - r).
16. Solve the Laplace's equation in three dimensions, ∇^2ϕ = 0, subject to appropriate boundary conditions.
17. Verify the truth value of the statement: For all positive integers n, if n^2 is even, then n is even.
18. Evaluate the integral ∫(e^x)/(x+1) dx.
19. Determine the probability of drawing two aces from a standard deck of 52 cards consecutively, without replacement.
20. Prove the Fundamental Theorem of Calculus.

These questions cover various branches of mathematics such as calculus, algebra, analysis, probability, and differential equations.
(Strictly not for Nonsensicals and Incomplete Equations.)​