University of Missouri Columbia Finding the Values Advanced Calculus Questions
Question Description
1. (5 pts) Suppose that f is six times continuously differentiable on[−1, 1]. Find all values of a, b for which the function f (x) = ax3 + bx4 +x6 has a local minimum at zero.
2. (5 pts) Suppose that f is a three times continuously differentiablefunction on [−2,2], f(0) = f′′(0) = 0, f(1) = 0. Which of the followingare possible?
(A) |f′′′(x)| ≤ 1 for every x ∈ (−2,2), and f′(0) = 51;(B) |f′′′(x)| ≤ 1 for every x ∈ (−2,2), and f′(0) = −1;
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(C) f′(0) = 5, and |f′′′(x)| ≤ 29 for every x ∈ (−2,2);(D) f′(0) = −5, and |f′′′(x)| ≤ 29 for every x ∈ (−2,2);
3. (5 pts) Suppose that a > 0.
(i) Prove that the function f(x) = ax is convex on R.(ii)Letx1,…xn ∈R,andx1+…+xn =0.Provethat
ax1 +ax2 +…+axn ≥n.
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