NIMCET 2012 — Mathematics PYQ

1. Analyze the Derivative

Let's find the first derivative of the function:

2. Determine Monotonicity

Observe the expression for $f^{\prime }(x)$:

• Since $x^4\ge 0$ for all real values of $x$, then $15x^4\ge 0$.

• Adding 15 makes the expression strictly positive: 15x^4 + 15 > 0 for all $x\in \mathbb{R}$.

Because f'(x) > 0 for all real $x$, the function $f(x)$ is strictly increasing over its entire domain.

3. Apply the Intermediate Value Theorem

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

• As $x\to \infty$, $f(x)\to \infty$.

Since the function is continuous and moves from negative infinity to positive infinity without ever "turning back" (because it is strictly increasing), it must cross the x-axis exactly once.

Conclusion:

The equation has exactly one real root.

Correct Option:

(c) 1