AIEEE 2008 (Offline)Question 1
How many real solutions does the equation $x^{7}+14x^{5}+16x^{3}+30x-560=0$ have?
Differentiation and its Application
Solution
Concept. A strictly increasing polynomial of odd degree has exactly one real root. Formulas used. (1) $f'>0$ everywhere $\Rightarrow$ at most one rootStep 1 - Derivative. $f'(x)=7x^6+70x^4+48x^2+30>0$. Step 2 - Existence. $f\to\pm\infty$ as $x\to\pm\infty$, so there is a root; by (1) there is only one. Answer: (B)