Conservative confidence interval for linear combination of parameters

Consider two parameters $alpha$ and $beta$ and let $C_alpha$, $C_beta$ be their $95%$ confidence intervals, respectively. Now, take the parameter $deltaequiv f(alpha,beta)$ where $f$ is a linear function of $(alpha,beta)$. Consider the interval $$Tequiv {cin mathbb{R}: exists (a,b) in C_alpha times C_beta text{ s.t. } c=f(a,b)}$$

1) Is the $95%$ confidence interval of $delta$ contained in $T$?

Instead, let $C_{alpha,beta}$ denote the joint $95%$ confidence region of $(alpha,beta)$. Consider the interval $$T_{joint}equiv {cin mathbb{R}: exists (a,b) in C_{alpha,beta} text{ s.t. } c=f(a,b)}$$

2) Is the $95%$ confidence interval of $delta$ contained in $T_{joint}$?

Remark: here I’m not assuming that the $C_alpha$, $C_beta$ come from two independent samples. They could be obtained using the same sample, or two dependent samples.