Which of the following logic statements are true for $mx^{2}+mx+4>0$

Question

Suppose that the formula $f(x, m)$ with two real-valued variables $m, x$,

$$f(x, m)=\big(mx^{2}+mx+4>0\big).$$ Which of the sentences are true :

a) $\exists m$ $\forall x $ $f(x, m)$

My answer: Consider $m=4$ so it is true for all $x$.

b) $\exists x$ $\forall m $ $f(x, m)$

My question : are my answers correct?

My answer : Consider $x=0$ It is true for all $m$.

Answer 1

The determinant of the polynomial is $m^2 - 16m$ and for $m=4$ we thus have a negative one, and so there are no (real) zeroes and the polynomial has constant sign. As its value at $0$ is positive indeed $f(x,m)$ always holds. So a) indeed is true. (but $m=1,2,3$ also work.)

And $f(0,m)$ is also always true. In short, you're right.