Question

The function $f$ defined by $f\left(x\right)=x^3-3x^2+5x+7,$ is:

• A. increasing in $R$
• B. decreasing in $R$
• C. decreasing in $\left(0,\infty \right)$ and increasing in $\left(-\infty ,0\right)$
• D. increasing in $\left(0,\infty \right)$ and decreasing in $\left(-\infty ,0\right)$

Answer 1

Answer: (A)

increasing in $R$

We have,

$\begin{array}{c}f\left(x\right)=x^3-3x^2+5x+7\\ f^{\prime }\left(x\right)=3x^2-6x+5\\ if{f}^{\prime }\left(x\right)>0\\ \Rightarrow 3x^2-6x+5>0\\ \Rightarrow 3\left(x^2-2x+\frac{5}{3}\right)>0\\ \Rightarrow 3\left(x^2-2x+1-1+\frac{5}{3}\right)>0\\ \Rightarrow 3{\left(x-1\right)}^2+2>0\end{array}$

It is increasing in $R$

Hence, this is the answer.