Application of DerivativeVery Hard
Question
The function f(x) = 
t(et - 1)(t - 1)(t - 2)3(t - 3)5dt minimum at x equals to
t(et - 1)(t - 1)(t - 2)3(t - 3)5dt minimum at x equals toOptions
A.0
B.1
C.2
D.3
Solution
f(x) = 
t(et - 1)(t - 1)(t - 2)3(t - 3)5dt
f′(x)
t(et - 1)(t - 1)(t - 2)3(t - 3)5dt
= x(ex -1)(x -1)(x - 2)3 (x - 3)5 × 1
For local minimum, f′(x) = 0
⇒ x = 0,1, 2,3.
Let f′(x) = g(x) = x(ex - 1)(x - 1)(x - 2)3 (x - 3)5
Using sign scheme rule,
This shows that f(x) has a local minimum at x = 1 and x = 3 and maximum at x = 2
Therefore, (b) and (d) are the correct answer.
t(et - 1)(t - 1)(t - 2)3(t - 3)5dt f′(x)
t(et - 1)(t - 1)(t - 2)3(t - 3)5dt = x(ex -1)(x -1)(x - 2)3 (x - 3)5 × 1
For local minimum, f′(x) = 0
⇒ x = 0,1, 2,3.
Let f′(x) = g(x) = x(ex - 1)(x - 1)(x - 2)3 (x - 3)5
Using sign scheme rule,
This shows that f(x) has a local minimum at x = 1 and x = 3 and maximum at x = 2
Therefore, (b) and (d) are the correct answer.
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