The function #f: f(x)=x^3+6x+2# is increasing when #x# belong to .......?

2 Answers
May 1, 2018

The given function is monotonically increasing when x belongs to the set of real numbers(#R#)

Explanation:

#f(-2) = -12#
#f(-1) = -5#
#f(0) = 2#
#f(1) = 9#

(Also, #fprime(x) = 3x^2 + 6#, which is always positive. So #f(x)#is strictly increasing.)

Clearly, f(x) increases as x increases.
graph{y = x^3 + 6x + 6 [-58.5, 58.5, -29.25, 29.3]}

May 1, 2018

The function is increasing #x in RR#

Explanation:

The function is

#f(x)=x^3+6x+2#

The derivative is

#f'(x)=3x^2+6#

The critical points are when #f'(x)=0#

As

#3x^2+6>0#

#AA x in RR, f'(x)>0#

Therefore,

The function is increasing #x in RR#

The second derivative is

#f''(x)=6x#

#f''(x)=0# when #x=0#

There is a point of inflection at #(0,2)#

graph{x^3+6x+2 [-2.48, 2.995, 0.721, 3.461]}