Math 408 Homework Set 1
(1) Find the local and global minimizers and maximizers of the following functions.
(a) f(x) = x2+ 2x
(b) f(x) = x2e−x2
(c) f(x) = x2+ sin x
(d) f(x) = x3−x
(2) Recall that a function f:Rn→Ris said to be differentiable at a point x∈Rnif there is a vector
g∈Rnsuch that
f(y) = f(x) + gT(y−x) + o(ky−xk).
The vector gis called the gradient of fat xand is denoted g=∇f(x). Note that, when defined, the
relation x7→ ∇f(x) is a mapping from Rnto Rn, i.e. ∇f:Rn→Rn. We say that fis continuously
differentiable at x∈Rnif the mapping ∇fis continuous at x. When fis continuously differentiable
at x∈Rn, then ∇f(x) is easily computed as the vector of partial derivatives of fat x, i.e.
∇f(x) =
∂f
∂x1(x)
∂f
∂x2(x)
.
.
.
∂f
∂xn(x)
.
Compute the gradient of the following functions.
(a) f(x) = x3
1+x3
2−3x1−15x2+ 25 : f:R2→R
(b) f(x) = x2
1+x2
2−sin(x1x2)f:R2→R
(c) f(x) = kxk2=Pn
j=1 x2
j:f:Rn→R
(d) f(x) = ekxk2
(e) f(x) = x1x2x3· · · xn:f:Rn→R
(f) f(x) = −log(x1x2x3· · · xn) for xj>0, j = 1,...n, and undefined otherwise: f:Rn→R.
Compute ∇f(x) for xj>0, j = 1,...n.
(3) Let Rn×ndenote the set of real n×nsquare matrices A function f:Rn→Ris said to be twice
differentiable at a point x∈Rnif is differentiable at xand there is a matrix H∈Rn×nsuch that
f(y) = f(x) + ∇f(x)T(y−x) + 1
2(y−x)TH(y−x) + o(ky−xk2).
The matrix His called the Hessian of fat xand is denoted ∇2f(x). Note that, when defined, the
relation x7→ ∇2f(x) is a mapping from Rnto Rn×n, i.e. ∇2f:Rn→Rn. We say that fis twice
continuously differentiable at x∈Rnif the mapping ∇2fis continuous at x. It can be shown that
if fis twice continuously differentiable at a point x∈Rn, then the matrix ∇2f(x) is symmetric,
i.e. ∇2f(x) = ∇2f(x)T, in which case ∇2f(x) is the matrix of second partial derivatives of fat x:
∇2f(x) =
∂2f
∂x1∂x1(x)∂2f
∂x2∂x1(x). . . ∂2f
∂xn∂x1(x)
∂2f
∂x1∂x2(x)∂2f
∂x2∂x2(x). . . ∂2f
∂xn∂x2(x)
.
.
..
.
.....
.
.
∂2f
∂x1∂xn(x)∂2f
∂x2∂xn(x). . . ∂2f
∂xn∂xn(x)
.
Compute the Hessian of the functions given in problem (2) above.
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