Question

The interval in which the function $x^3$ increases less rapidly than $6x^2+15x+5$

• A. $\left(-\infty ,-1\right)$
• B. $\left(-5,1\right)$
• C. $\left(-1,5\right)$
• D. $\left(5,\infty \right)$

Answer 1

The correct option is C

$\left(-1,5\right)$

Explanation for correct option:

Find the interval:

Given that the function $x^3$ increases less rapidly than $6x^2+15x+5$.

We know that for a function $f\left(x\right)$ increase rate is determined by $\frac{df\left(x\right)}{dx}$.

Therefore,

$$\begin{gathered} \frac{d{x}^3}{dx}<\frac{d\left(6x^2+15x+5\right)}{dx}\mathbf{}\mathbf{[}\mathbf{\because }\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}{\mathbf{x}}^{\mathbf{n}}\mathbf{=}\mathbf{n}\mathbf{}{\mathbf{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{]} \\ \Rightarrow 3x^2<12x+15 \\ \Rightarrow 3x^2-12x-15<0 \\ \Rightarrow x^2-4x-5<0 \\ \Rightarrow x^2-5x+x-5<0 \\ \Rightarrow x\left(x-5\right)+\left(x-5\right)<0 \\ \Rightarrow \left(x-5\right)\left(x+1\right)<0 \\ x\in \left(-1,5\right) \end{gathered}$$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{C}\right)$