Question

Find the point(s) where the function f(x) = $x^3$ - 3x + 2 is increasing

• A. x = 0
• B. x = 1
• C. x = 2
• D. None of these

Answer 1

The correct option is C

x = 2

To decide the nature of f(x) about a point, we will take the derivative and observe the sign. If we get f’(x) equal to zero at the point, then we should further investigate the sign of f’(x) in the neighborhood to decide the nature of f(x).

f’(x) = 3 $x^2$ - 3 .

f’(0) = -3 < 0

f’(0) < 0 means at x = 0, the function f(x) is decreasing.

f’(1) = 0

Since f’(1) is zero, we will look at the sign of f’(x) for x = 1- h and 1+h. If they are positive, we can say f(x) is increasing about x = 1 and if they are negative, we can say that f(x) is decreasing about x =1. In case the signs are opposite, x =1 is not a point of monotonicity

f’(1+h )= 3 $(1+h{)}^2$ -1) = 3 $(h^2+2h)$ > 0

f’(1-h) = 3 $(1-h{)}^2$ -1) =3 $(h^2+2h)$ = 3h ( h-2) < 0 , because h is very small compared to 2

Since the sign of f’(x) is opposite on both sides of 1, we can say it is not point of monotonicity

Now, f’(2) = 9

Since f’(2) is positive, we can say f(x) is increasing about the point x = 2