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Calculus Level 2

$f(x) = 2x^3 -15x^2 + 36x -27$

Let $a$ and $b$ be the distinct roots of the function above. If $(c , d)$ is the domain, where $f(x)$ is increasing within $(a , b)$ , find $c\times d$ .

The answer is 3.00.

1 solution

Worranat Pakornrat
Mar 2, 2016

By factorization, we can rewrite f(x) as: $f(x) = 2x^3 -15x^2 + 36x -27 = (2x - 3)(x - 3)(x - 3)$ .

Therefore, the roots are $1.5$ and $3$ for f(x); $a = 1.5$ and $b = 3$ .

Now by differentiation: $\begin{aligned} f'(x) &= 6x^2 - 30x +36 \\ &= 6(x^2 - 5x + 6) \\ &= 6(x - 2)(x - 3) \end{aligned}$

It is obvious that $f'(x) = 0$ at $x = 2$ and $x = 3$ , and within the domain $(2 , 3)$ , $f'(x) < 0$ . That means the graph will be increasing when $x < 2$ or $x > 3$ .

Therefore, within the roots domain $(1.5 , 3)$ , the graph's increasing when $x < 2$ or $x \in (1.5 , 2)$ .

Thus, $c = 1.5$ and $d = 2$ . As a result, $c\times d = 1.5\times 2 = \boxed{3}$ .

Moderator note:

Good explanation.

I made it obvious in the problem that $a \neq b$ .

Rise within Roots

The answer is 3.00.

1 solution

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