Question

Find the range of values of x for which each of the following functions is decreasing
(x^2)/2-5x+1
x^3-(5/2)x^2-2x+1

Answer 1

To find the range of values of x for which a function is decreasing, we need to find the interval(s) where the derivative of the function is negative.

For the first function, f(x) = (x^2)/2 - 5x + 1, let's find its derivative:
f'(x) = (2x)/2 - 5 = x - 5

To find when f'(x) < 0:
x - 5 < 0
x < 5

So, the first function is decreasing for x < 5.

For the second function, f(x) = x^3 - (5/2)x^2 - 2x + 1, let's find its derivative:
f'(x) = 3x^2 - 5x - 2

To find when f'(x) < 0, we can use the quadratic formula:
x = (-(-5) ± sqrt((-5)^2 - 4(3)(-2))) / (2(3))
x = (5 ± sqrt(25 + 24)) / 6
x = (5 ± sqrt(49)) / 6
x = (5 ± 7) / 6

So, x = 2/3 or x = 4.

Therefore, the second function is decreasing for x < 2/3 and x > 4.