Question

Let $f: \mathbb{R}^4 \to \mathbb{R}$ be defined by $f(x) = x^T A x$, where A is a $4 \times 4$ matrix with real entries and $x^T$ denotes the transpose of $x$. The gradient of $f$ at a point $x_0$ necessarily is (a) $2 A x_0$ (b) $A x_0 + A^T x_0$ (c) $2 A^T x_0$ (d) $A x_0$

          Let $f: \mathbb{R}^4 \to \mathbb{R}$ be defined by $f(x) = x^T A x$, where A is a $4 \times 4$ matrix with real entries and $x^T$ denotes the transpose of $x$. The gradient of $f$ at a point $x_0$ necessarily is
(a) $2 A x_0$
(b) $A x_0 + A^T x_0$
(c) $2 A^T x_0$
(d) $A x_0$

Let f: ℝ^4 →ℝ be defined by f(x) = x^T A x, where A is a 4 × 4 matrix with real entries and x^T denotes the transpose of x. The gradient of f at a point x0 necessarily is
(a) 2 A x0
(b) A x0 + A^T x0
(c) 2 A^T x0
(d) A x0

Added by Irene G.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Danielle Fairburn

05/11/2024

Step 1

This is a quadratic form where $ x $ is a vector in $\mathbb{R}^4$ and $ A $ is a $4 \times 4$ matrix. Show more…

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let $f: \mathbb{R}^4 \rightarrow \mathbb{R}$ be defined by $f(x) = x^tAx$, where A is 4x4 matrix with real entries and $x^t$ denotes the transpose of x. The gradient of f at a point $x_0$ necessarily is (a) $2Ax_0$ (b) $Ax_0 + A^tx_0$ (c) $2A^tx_0$ (d) $Ax_0$