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Let f (x) = x3−6x2+9x+18, then f (x) is strict decreasing in
- a)[3,∞)
- b)[−∞,1)
- c)[1, 3]
- d)(−∞,1]
Correct answer is option 'C'. Can you explain this answer?
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Let f (x) =x3−6x2+9x+18,then f (x) is strict decreasing ina)[3,&...
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Let f (x) =x3−6x2+9x+18,then f (x) is strict decreasing ina)[3,&...
Explanation:
Given function:
f(x) = x^3 - 6x^2 + 9x - 18
Finding the derivative:
To determine whether the function is strictly decreasing, we need to find its derivative.
f'(x) = 3x^2 - 12x + 9
f'(x) = 3(x^2 - 4x + 3)
f'(x) = 3(x - 3)(x - 1)
Finding critical points:
Setting f'(x) = 0 to find the critical points:
3(x - 3)(x - 1) = 0
x = 3 or x = 1
Testing intervals:
Now, we need to test the intervals around the critical points to determine the function's behavior.
Interval [1, 3]:
Pick a value between 1 and 3, say x = 2
f'(2) = 3(2 - 3)(2 - 1) = -3 < />
Since the derivative is negative in this interval, the function is strictly decreasing in [1, 3].
Conclusion:
Therefore, the function f(x) = x^3 - 6x^2 + 9x - 18 is strictly decreasing in the interval [1, 3].
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