Let $f:\mathbb{R}\rightarrow [0,\infty)$ be a continuous function. Then which of the following is not true?
(a) There exist a $x\in \mathbb{R}$ such that $f(x)=\frac{f(0)+f(1)}{2}$
(b) There exist a $x\in \mathbb{R}$ such that $f(x)=\sqrt{f(-1)f(1)}$
(c) There exist a $x\in \mathbb{R}$ such that $f(x)=\int_{-1}^{1}f(t)dt$
(d) There exist a $x\in \mathbb{R}$ such that $f(x)=\int_{0}^{1}f(t)dt$
Attempt
For $(a)$ take $f(x)=5$, for $(b)$ take $f(x)=1$. So a and b can be false in general.
I need some hints for $c$ and $d$. Can I use fundamental theorem of Calculus here?
Assertion (c) is false in general. Just take $f(x)=1$. All the others are a consequence of the intermediate value theorem.