largest value of A

Question

 Find the largest value of A such that the function 𝑓(𝑥)=𝑥3+1𝑥2−6𝑥+6 is decreasing for all x in the interval (0,𝐴)

Answer 1

I'm going to assume that you did not make an error in writing the initial function (you wrote 1x² which is a little weird).

So if you want to ensure a function is decreasing, then you want to make its derivative is negative.

Using the power rule, f '(x) = 3x² + 2x - 6

To determine where 3x² + 2x - 6 is negative, find the zeros:

Using the quadratic formula:

x = (-b±√(b²-4ac))/(2a)

x = (-2±√(4+72))/6

x = (-2 ± 2√19)/6

x = (-1 ± √19)/3

This is a quadratic with positive leading coefficient so it will open upwards and will be negative between the zeros. One zero (-1 - √19)/6 is a negative number so since we are looking for an interval from (0, A) we can ignore this one in terms of "A". So the interval that ensures f(x) is decreasing would mean A = (-1 + √19)/3

Answer 2

The function decreasing means having a df/dx < 0

df/dx = 3x² + 2x - 6

find the zeroes using quadratic formula. The positive zero is (-2 +sqrt(76))/6 = A

You can check the sign from 0 to that number by inputting 0 or 1 and getting df/dx < 0. The df/dx won't change sign until greater than A or less than the other root (which is <0)

Please consider a tutor. Take care.