JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Application of Derivatives

Let $f(x)=x^7+5x^3+3x+1$.

Find the derivative:


Since $f^{\prime }(x)>0$ everywhere, $f(x)$ is strictly increasing on $\mathbb{R}$.

A strictly increasing function can cross the x-axis at most once. Since:

• As $x\to -\infty$, $f(x)\to -\infty$
• As $x\to \infty$, $f(x)\to \infty$

By the Intermediate Value Theorem, there is exactly one real root.