## Elementary Linear Algebra 7th Edition

(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.
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 $$|\langle f,g \rangle|=1\leq\| f \|\| g \|=\sqrt{\frac{1}{3}}\sqrt{\frac{16}{5} }$$