Explanation
Step 1: Find the derivative of f(x)
Given:
f(x)=3x3−25x2+6x+7
The derivative is:
f′(x)=dxd(3x3−25x2+6x+7)
f′(x)=x2−5x+6
Step 2: Find the critical points
Set the derivative equal to zero to find the critical points:
x2−5x+6=0
(x−2)(x−3)=0
Thus, the critical points are x=2 and x=3.
Step 3: Determine the intervals of increase
The function increases when f'(x) > 0:
x^2 - 5x + 6 > 0
(x - 2)(x - 3) > 0
Testing the intervals defined by the critical points x=2 and x=3:
For x < 2 (e.g., x=0): (0-2)(0-3) = (-2)(-3) = 6 > 0 (Increasing)
For 2 < x < 3 (e.g., x=2.5): (2.5-2)(2.5-3) = (0.5)(-0.5) = -0.25 < 0 (Decreasing)
For x > 3 (e.g., x=4): (4-2)(4-3) = (2)(1) = 2 > 0 (Increasing)
Therefore, the function increases on the interval T = \{x < 2\} \cup \{x > 3\}.
The correct option is (b).
