Is #f(x)=-7x^3+x^2-2x-1# increasing or decreasing at #x=-2#?

2 Answers
Jun 3, 2018

See explanation.

Explanation:

To find if a function is increasing or decreasing at a given point we have to calculate its derivative at the point.

The function is:

#f(x)=-7x^3+x^2-2x-1#

so its derivative is:

#f^'(x)=-21x^2+2x-2#

The value of derivative at #x_0=-2# is:

#f^'(2)=-21*(-2)^2+2*(-2)-2=-84-4-2=-90#

The value of the derivative is less than zero, so the function is decreasing.

Jun 3, 2018

Either drawing the graph or taking the derivate #f'(x)=-21x^2+2x-2# in #x=-2# we find that the slope is negative (#f'(-2)=-90#). We can, therefore, conclude that #f(x)# is decreasing at #x=-2#

Explanation:

Let's start by drawing the graph in Geogebra (an excellent, free geometry program):
enter image source here
Looking at it , it is quite clear that #f(x)# is decreasing at #x=-2#

We can also find this without drawing the graph by taking the derivate, which gives the slope of the tangent in each value of x:
#f'(x)=-7*3x^2+2x-2=-21x^2+2x-2#
Insert the value #x=-2#:
#f'(-2)=-21(-2)^2+2(-2)-2=-84-4-2=-90#

As #f'(-2)=-90<0#, we can conclude that #f(-2)# is decreasing in -2.