How many real solutions does the equation $${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$       have ?

Solution :
$$\eqalign{ & {\text{Let}}\,{\text{f}}\left( x \right) = \,{x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0 \cr & \Rightarrow f'\left( x \right) = 7{x^6} + 70{x^4} + 48{x^2} + 30 > 0,\forall \,x \in R \cr & \Rightarrow f\,{\text{is}}\,{\text{an}}\,{\text{increasing}}\,{\text{function}}\,{\text{on}}\,R \cr & {\text{Also}}\,\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = \infty \,{\text{and}}\,\mathop {\lim }\limits_{x \to - \infty } f\left( x \right) = - \infty \cr & \Rightarrow \,{\text{The}}\,{\text{curve}}\,y = f\left( x \right)\,{\text{crosses}}\,x\,{\text{ - axis}}\,{\text{only}}\,{\text{once}}. \cr & \therefore f\left( x \right) = 0\,{\text{has}}\,{\text{exactly}}\,{\text{one}}\,{\text{real}}\,{\text{root}}. \cr} $$