Question

Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 5 + 54x − 2x3, [0, 4]

Answer 1

Answer:

The absolute maximum and minimum of $f(x) = 5 + 54\cdot x -2\cdot x^{3}$ on $[0,4]$ are 113 and 5.

Step-by-step explanation:

Let be $f(x) = 5 + 54\cdot x -2\cdot x^{3}$, the first and second derivatives of the function are, respectively:

$f'=54-6\cdot x^{2}$

$f''=-12\cdot x$

Now, let equalize the first derivative to zero and solve the resulting expression:

$54-6\cdot x^{2} = 0$

$x^{2} = 9$

$x =\pm 3$

According to the given interval, only $x=3$ is a valid outcome. Lastly, this is evaluated in the second derivative expression:

$f''=-12\cdot(3)$

$f'' = -36$

$x = 3$ leads to an absolute maximum.

$f(3) = 5 + 54\cdot (3) -2\cdot (3)^{3}$

$f(3) = 113$

The absolute minimum is determined by evaluating at each extreme of the interval:

$x = 0$

$f(0) = 5 + 54\cdot (0) -2\cdot (0)^{3}$

$f(0) = 5$

$x = 4$

$f(4) = 5 + 54\cdot (4) -2\cdot (4)^{3}$

$f(4) = 93$

The absolute maximum and minimum of $f(x) = 5 + 54\cdot x -2\cdot x^{3}$ on $[0,4]$ are 113 and 5.