Answer
(a) The maximum value is approximately $2.20$
The minimum value is approximately $1.80$
(b) The maximum value is $2.21$
The minimum value is approximately $1.81$
Work Step by Step
(a) $f(x) = x^5-x^3+2$
We can use the zoom function on a graphing calculator to estimate the maximum and minimum values of the function in the interval $[-1,1]$
The maximum value is approximately $2.20$
The minimum value is approximately $1.80$
(b) $f(x) = x^5-x^3+2$
$f'(x) = 5x^4-3x^2 = 0$
$x^2(5x^2-3) = 0$
$x=0$ or $5x^2 = 3$
$x=0$ or $x = \pm \sqrt{\frac{3}{5}}$
When $x = -\sqrt{\frac{3}{5}}$:
$(-\sqrt{\frac{3}{5}})^5-(-\sqrt{\frac{3}{5}})^3+2 = 2.19$
The maximum value is $2.21$
When $x = \sqrt{\frac{3}{5}}$:
$(\sqrt{\frac{3}{5}})^5-(\sqrt{\frac{3}{5}})^3+2 = 1.81$
The minimum value is approximately $1.81$
