First slide

Monotonicity

Question

$f (x) = (x - 8)^{4} (x - 9)^{5}, 0 \leq x \leq 10,$ monotonically decreases in

Moderate

A

$(\frac{76}{9}, 10]$
B

$(8, \frac{76}{9})$
C

$[0, 8)$
D

$(\frac{76}{9}, - 10]$

Solution

$f (x) = (x - 8)^{4} (x - 9)^{5}, 0 \leq x \leq 10 ∴ f^{'} (x) = 4 (x - 8)^{3} (x - 9)^{5} + 5 (x - 9)^{4} (x - 8)^{4} = (x - 8)^{3} (x - 9)^{4} [4 (x - 9) + 5 (x - 8)] = 9 (x - 8)^{3} (x - 9)^{4} (x - \frac{76}{9})$
Sign scheme of f'(x) is as follows:

$f^{'} (x) < 0 if x \in (8, \frac{76}{9})$ ,i.e., f(x) decreases if $x \in (8, \frac{76}{9})$ .

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