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Let $f(x): \mathbb{R}^n \to \mathbb{R}$ be a real-valued twice continuously differentiable function and $n>1$. I define the function $g(x) = f(x) + x^{\top} A x$ where $A$ is random matrix (say entries i.i.d from uniform distribution [-1,1]). Can we say that the Hessian of $g$ is invertible for all $x$ with probability one?

Let $f(x): \mathbb{R}^n \to \mathbb{R}$ be a real-valued twice continuously differentiable function. I define the function $g(x) = f(x) + x^{\top} A x$ where $A$ is random matrix (say entries i.i.d from uniform distribution [-1,1]). Can we say that the Hessian of $g$ is invertible for all $x$ with probability one?

Let $f(x): \mathbb{R}^n \to \mathbb{R}$ be a real-valued twice continuously differentiable function and $n>1$. I define the function $g(x) = f(x) + x^{\top} A x$ where $A$ is random matrix (say entries i.i.d from uniform distribution [-1,1]). Can we say that the Hessian of $g$ is invertible for all $x$ with probability one?

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Let $f(x): \mathbb{R}^n \to \mathbb{R}$ be a real-valued twice continuously differentiable function. I define the function $f'(x) = f(x) + x^{\top} A x$$g(x) = f(x) + x^{\top} A x$ where $A$ is random matrix (say entries i.i.d from uniform distribution [-1,1]). Can we say that the Hessian of $f$$g$ is invertible for all $x$ with probability one?

Let $f(x): \mathbb{R}^n \to \mathbb{R}$ be a real-valued twice continuously differentiable function. I define the function $f'(x) = f(x) + x^{\top} A x$ where $A$ is random matrix (say entries i.i.d from uniform distribution [-1,1]). Can we say that the Hessian of $f$ is invertible for all $x$ with probability one?

Let $f(x): \mathbb{R}^n \to \mathbb{R}$ be a real-valued twice continuously differentiable function. I define the function $g(x) = f(x) + x^{\top} A x$ where $A$ is random matrix (say entries i.i.d from uniform distribution [-1,1]). Can we say that the Hessian of $g$ is invertible for all $x$ with probability one?

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Question on Hessian of a function (probability question)

Let $f(x): \mathbb{R}^n \to \mathbb{R}$ be a real-valued twice continuously differentiable function. I define the function $f'(x) = f(x) + x^{\top} A x$ where $A$ is random matrix (say entries i.i.d from uniform distribution [-1,1]). Can we say that the Hessian of $f$ is invertible for all $x$ with probability one?