Answer
(a) $\langle f, g\rangle =1,$
(b) Since $\langle f, g\rangle=1\neq 0$, then $f$ and $g$ are not orthogonal.
(c) see the details.
Work Step by Step
Let $f(x)=x, \quad g(x)=4x^2.$, $\langle f, g\rangle=\int_{0}^1 f(x)g(x) d x$, we have
(a) $\langle f, g\rangle=\int_{0}^1 4x^3d x= \left[x^4\right]_{0}^1=1,$
(b) Since $\langle f, g\rangle=1\neq 0$, then $f$ and $g$ are not orthogonal.
(c) To verify the Cauchy-Schwarz Inequality, we have
$$\langle f,f\rangle=\int_{0}^1x^2 d x=frac{1}{3}\left[x^3\right]_{0}^1=\frac{1}{3},$$
$$\langle g, g\rangle=\int_{0}^1 16x^4 d x=\frac{16}{5}\left[x^5\right]_{0}^1=\frac{16}{5} .$$
Since for any $f$ we have $\| f \| =\sqrt{\langle f, f\rangle} $, then we have the Cauchy-Schwarz Inequality
$$|\langle f,g \rangle|=1\leq\| f \|\| g \|=\sqrt{\frac{1}{3}}\sqrt{\frac{16}{5} }$$
