If $f(x)=\frac{x^3}{3}-\frac{5 x^2}{2}+6 x+7$ increases in the interval T and decreases in the interval S , then which one of the following is correct?

• Correct answer
  A

  $\mathrm{T}=(-\infty, 2) \cup(3, \infty)$ and $\mathrm{S}=(2,3)$
• B

  $\mathrm{T}=\phi$ and $\mathrm{S}=(-\infty, \infty)$
• C

  $\mathrm{T}=(-\infty, \infty)$ and $\mathrm{S}=\phi$
• D

  $\mathrm{T}=(2,3)$ and $\mathrm{S}=(-\infty, 2) \cup(3, \infty)$

Solution

$\begin{aligned} & \text { } f(x)=\frac{x^3}{3}-\frac{5 x^2}{2}+6 x+7 \\ & f^{\prime}(x)=\frac{3 x^2}{3}-\frac{5 \cdot 2 \cdot x}{2}+6 \\ & \Rightarrow f^{\prime}(x)=x^2-5 x+6 \\ & \Rightarrow f^{\prime}(x)=(x-3)(x-2)\end{aligned}$
$\begin{aligned} & \therefore \quad T \in(-\infty, 2) \cup(3, \infty) \\ & \text {and } S \in(2,3)\end{aligned}$

Hence, the correct answer is option: 'A'.