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Magazine Problem 49
Use Theorem 11.3.5(c) or, where applicable, Exercise 11.1.42(b), to find the mixed Fourier cosine series of \(f\) on \([0, L]\). $$ f(x)=x^{4}-4 L x^{3}+6 L^{2} x^{2}-3 L^{4} $$
Problem 51
Use Theorem \(11.3 .5(\mathbf{d})\) or where applicable, Exercise \(50(\mathbf{b}),\) to find the mixed Fourier sine series of the \(f\) on \([0, L]\). $$ f(x)=x(2 L-x) $$
Problem 52
Use Theorem \(11.3 .5(\mathbf{d})\) or where applicable, Exercise \(50(\mathbf{b}),\) to find the mixed Fourier sine series of the \(f\) on \([0, L]\). $$ f(x)=x^{2}(3 L-2 x) $$
Problem 53
Use Theorem \(11.3 .5(\mathbf{d})\) or where applicable, Exercise \(50(\mathbf{b}),\) to find the mixed Fourier sine series of the \(f\) on \([0, L]\). $$ f(x)=(x-L)^{3}+L^{3} $$
Problem 54
Use Theorem \(11.3 .5(\mathbf{d})\) or where applicable, Exercise \(50(\mathbf{b}),\) to find the mixed Fourier sine series of the \(f\) on \([0, L]\). $$ f(x)=x\left(x^{2}-3 L^{2}\right) $$
Problem 55
Use Theorem \(11.3 .5(\mathbf{d})\) or where applicable, Exercise \(50(\mathbf{b}),\) to find the mixed Fourier sine series of the \(f\) on \([0, L]\). $$ f(x)=x^{3}(3 x-4 L) $$
Problem 56
Use Theorem \(11.3 .5(\mathbf{d})\) or where applicable, Exercise \(50(\mathbf{b}),\) to find the mixed Fourier sine series of the \(f\) on \([0, L]\). $$ f(x)=x\left(x^{3}-2 L x^{2}+2 L^{3}\right) $$
Problem 58
Show that the mixed Fourier sine series of \(f\) on \([0, L]\) is the restriction
to \([0, L]\) of the Fourier sine series of
$$
f_{4}(x)=\left\\{\begin{array}{cc}
f(x), & 0 \leq x \leq L \\
f(2 L-x), & L
