Question

Is `f(x)=(x-3)(x+11)(x-7)` increasing or decreasing at `x=-1`?

Answer 1

f(x) is decreasing at x = -1

Explanation:

distribute the brackets before differentiating , is probably ' better' than using the 'product rule', in this case.

(x - 3 )(x + 11 ) = `x^2+ 8x - 33`

and `(x^2 + 8x -33 )(x - 7 )`

`= x^3 - 7x^2 + 8x^2 - 56x - 33x + 231`
`= x^3 + x^2 - 89x + 231`

to test whether the function is increasing/decreasing, require to check the value of f'(-1)

• If f'(-1) > 0 then f(x) is increasing at x = -1

• If f'(-1) < 0 then f(x) is decreasing at x = -1

hence f(x) = `x^3 + x^2 - 89x + 231`

so f'(x) `= 3x^2 + 2x -89`

and f'(-1)`= 3(-1)^2 + 2(-1) - 89 = 3 - 2 - 89 = - 88 < 0`

hence f(x) is decreasing at x = - 1