What is the F distribution used for in general linear models? That bascially says:

Diet influences cholestrol level. The standard model below related to cholestrol level to the content of saturated fats  in the diet as
$$Y = \beta_0 + \beta_1 x + \epsilon $$
where $x$ describes consumption of saturated fats. It is believed that eating fruits can lower cholestrol level. Define the new multiple regression model including the fruit influence:

So I said it was
$$Y = \beta_0 + \beta_1x + \beta_2 f + \epsilon$$
Where $f$ is fruit consumption. Then, I was asked

Propose a test to check if eating fruit affects cholestrol level

So I said, I will use the F-test where $H_0$ will be that fruit doesn't affect cholestrol levels. However in the answers, it says they want to test if $\beta_2 = 0$ by comparing the residual sum of squares for the null model ($\beta_2 = 0$) with the sum of squares given to me earlier on in the question.
I understand that this comparing thing is basically just doing the F-test, but why do they look at $\beta_2 = 0$?. Is it because the F - test is used, in GLMs, to see if there are any constraints on the estimator terms?
 A: It is simpler to talk about this issue in the context of Gaussian multiple linear regression, which is the context where you get exact T-distributions and F-distributions.  In the broader context of GLMs you get asymptotic results with those distributions under broad conditions, but it comes with some caveats.  I am going to talk in the context of Gaussian linear regression first, to get to the root of the issue.
If you are testing an individual coefficient in Gaussian linear regression, then the T-test on a single coefficient is equivalent to the F-test for the singleton subset composed only of that coefficient.  This reflects the fact that the  square of a T random variable is an F random variable with one numerator degree-of-freedom.  So this means that for a testing whether a single coefficient is non-zero, the standard coefficient test (using the T distribution) is equivalent to the partial goodness-of-fit test (using the F distribution).  Now, in the broader context of a GLM, these two tests are still equivalent, but now both of them hold as approximate tests using asymptotic distributions (instead of exact distributions) for the test statistics.
In the present case, both the T-test or the partial F-test would be testing the null hypothesis $H_0: \beta_2 = 0$ against the alternative hypothesis $H_\text{A}: \beta_2 \neq 0$.  This is because, under the null hypothesis, the regression term containing the fruit consumption $f$ drops out of the equation and so there is no statistical relationship between fruit consumption and cholesterol level.
