JEE Main · 2008 · 27 Apr 2008 · Offline · Application of Derivatives

Differentiating $f(x)=x^7+14x^5+16x^3+30x-560$ gives $f^{\prime }(x)=7x^6+70x^4+48x^2+30$, a sum of terms that are never negative with a constant that keeps it strictly positive for every real $x$. A function with an everywhere-positive derivative is strictly increasing and can cross the $x$-axis at most once; since it runs from $-\infty$ to $+\infty$ it does so exactly once, so the equation has a single real solution.