Is the product of a concave function and the exponential function concave?

Question

Let $g(\theta) = \sum_{i=1}^n f(x_i^T\theta - y_i)$ be concave function, where $\theta \in {\mathbb R^p}$ and $(y_i,x_i)$ are fixed (scalar and vector, respectively). Is

$h(\theta)= \sum_{i=1}^n f(x_i^T\theta - y_i)\exp(x_i^T\theta)$

concave too?

Answer 1

Not necessarily. For example, when $f$ is linear and attains positive value over the domain, then $h$ is convex.