Let M and m be respectively the absolute maximum and the absolute minimum values of the fu…

Question

Let M and m be respectively the absolute maximum and the absolute minimum values of the function,  f(x) = 2x3 - 9x2 + 12x + 5 in the interval [0, 3]. Then M -m is equal to :

Accepted Answer

To determine the absolute maximum (M) and absolute minimum (m) of the function $ f(x) = 2x^3 - 9x^2 + 12x + 5 $ over the interval $[0, 3]$, we need to examine its critical points and endpoints.

First, we find the derivative of the function, $ f'(x) $, to locate the critical points:

$$ f'(x) = \frac{d}{dx} (2x^3 - 9x^2 + 12x + 5) = 6x^2 - 18x + 12 $$

Next, we set the derivative equal to zero to find the critical points:

$6x^2 - 18x + 12 = 0$

Simplifying this equation by dividing by 6:

$x^2 - 3x + 2 = 0$

We solve this quadratic equation using the factorization method:

$(x - 1)(x - 2) = 0$

So, the critical points are:

$$ x = 1 \quad \text{and} \quad x = 2 $$

We now evaluate the function $ f(x) $ at the critical points and at the endpoints of the interval [0, 3]:

1. At $ x = 0 $:

$f(0) = 2(0)^3 - 9(0)^2 + 12(0) + 5 = 5$

2. At $ x = 1 $:

$$ f(1) = 2(1)^3 - 9(1)^2 + 12(1) + 5 = 2 - 9 + 12 + 5 = 10 $$

3. At $ x = 2 $:

$$ f(2) = 2(2)^3 - 9(2)^2 + 12(2) + 5 = 16 - 36 + 24 + 5 = 9 $$

4. At $ x = 3 $:

$$ f(3) = 2(3)^3 - 9(3)^2 + 12(3) + 5 = 54 - 81 + 36 + 5 = 14 $$

We now identify the absolute maximum (M) and absolute minimum (m) from the above values:

Maximum value $ M = 14 $ at $ x = 3 $

Minimum value $ m = 5 $ at $ x = 0 $

Thus, the difference $ M - m $ is:

$M - m = 14 - 5 = 9$

Therefore, the correct answer is:

Option B: 9

Solutions · 1