What is the minimum value of the expression f(x) = x^3 - 5x + 8

Now we have to find the minimum value of f(x) = x^3 - 5x
+8.

For that we start with finding the derivative of f(x) =
x^3 - 5x +8

f’(x) = [x^3 + 5x +8]’ = 3x^2 +
5

Now equate this to
0

=> 3x^2 - 5 =
0

=> x^2 =
5/3

=> x = -sqrt [5/3] or +sqrt
[5/3]

Now take the second derivative of
f(x)

f’’(x) = 6x

for x= -sqrt
[5/3], 6x = -6*sqrt [5/3] which is negative, therefore the maximum value is at x= -sqrt
[5/3]

for x= +sqrt [5/3] , 6x = +6*sqrt [5/3] which is
positive, therefore the minimum value is at x = +sqrt
[5/3]

The minimum value is: f[(+sqrt [5/3])] = (+sqrt
[5/3])^3 – 5*(+sqrt [5/3]) +8 = 3.69

The
required result is 3.69