Find the minimum of g(t)=t^3-25t+6 on the interval [-2.5, 5].

We have to find the minimum g(t)=t^3-25t+6 on the interval
[-2.5, 5].

First, we find the derivative of g(t). That is
equated to 0 to solve for x. The value of the second derivative is positive where the
function forms a
minimum.

g(t)=t^3-25t+6

g'(t)
= 3t^2 - 25

3t^2 - 25 =
0

=> t^2 =
25/3

=> t = -sqrt (25/3) and +sqrt
(25/3)

For these + sqrt(25/3) lies in the given
interval.

g''(t) for t = sqrt(25/3) is positive indicating
that we have a minimum.

The minimum value is [sqrt
(25/3)]^3 - 25*sqrt(25/3) + 6

=> (25/3)*(sqrt(25/3)
- 25*sqrt(25/3) + 6

=> sqrt(25/3)[25/3 - 25] +
6

=> (-50/3)*sqrt(25/3) +
6

=> -250/3*sqrt 3 +
6

The required minimum of g(t)= t^3-25t+6 on
the interval [-2.5, 5] is -250/(3*sqrt 3) + 6