×

Hello! Socratic's Terms of Service and Privacy Policy have been updated, which will be automatically effective on October 6, 2018. Please contact hello@socratic.com with any questions.

For what values of x is #f(x)= -9x^3 + 4 x^2 + 7x -2 # concave or convex?

2 Answers
Feb 18, 2017

Answer:

#f(x)# is convex (concave up) for #x in ("–"oo," "4/27)#.
#f(x)# is concave (concave down) for #x in (4/27," "oo)#.

Explanation:

A function is concave (or concave down) where its derivative is decreasing. Graphically, a concave region looks like a cave (or cut from a cave shape). (Quick example: the function #f(x)="–"x^2# is concave everywhere, since it never curves upward.)

A function is convex (or concave up) where its derivative is increasing. Graphically, a convex region looks like a V (or cut from a V shape). (The function #f(x)=x^2# is convex everywhere.)

To find these regions, we need to analyze the behaviour of the function's derivative. Hence, we need to find the derivative of the derivative: #f''(x)#.

Given #f(x)="–"9x^3+4x^2+7x-2#, we have

#f'(x)="–"27x^2+8x+7#

by the power rule, and so the derivative of this is

#f''(x)="–"54x+8#.

Just like how we know #f(x)# is increasing when #f'(x)>0#, we know #f'(x)# is increasing when #f''(x)>0#. Using this, we find:

#f''(x)="–"54x+8 > 0#

#=>"                 –"54x>"–8"#

#=>"                       "x<8/54=4/27#.

So #f(x)# is convex (concave up) for #x in ("–"oo," "4/27)#. This means #x=4/27# is an inflection point—a point at which the direction of concavity changes.

Finding where #f(x)# is concave (concave down) means finding where #f''(x)<0#, but that's equivalent to swapping #># signs for#<# signs from when we found #f''(x) > 0#. So our answer will be #f(x)# is concave for #x in (4/27," "oo)#.

Here's a graph of the function, with the inflection point circled.
graph{(-9x^3+4x^2+7x-2-y)((x-4/27)^2+(y+0.9)^2/36-0.0025)=0 [-2.1, 2.1, -6, 6]}

As we come from #"–"oo#, the slope (of the tangent line) of the graph is increasing, right up until the inflection point, where the slope begins to decrease.

Notice how the function is V-shaped to the left of the inflection point, and cave-shaped to its right. That's an easy way to remember the difference between concave and convex: concave is like a cave; convex is like a V.

Feb 18, 2017

Answer:

#f(x)# is convex when #x in ]-oo,0.678]# and concave when #x in [-0.382,+oo[#

Explanation:

We calculate the first derivative and build a chart of variations

#f(x)=-9x^3+4x^2+7x-2#

#f'(x)=-27x^2+8x+7#

To determine the critical points, we solve the equation

#-27x^2+8x+7=0#

#Delta=8^2-4*(-27)*(7)=820#

As, #Delta>0#, there are 2 real roots

#x_1=(-8+sqrt820)/(2*-27)=-0.382#

#x_2=(-8-sqrt820)/(2*-27)=0.678#

The chart of variations is

#color(white)(aaaa)##x##color(white)(aaaaaa)##-oo##color(white)(aaa)##-0.382##color(white)(aaaa)##0.678##color(white)(aaaa)##+oo#

#color(white)(aaaa)##x-x_1##color(white)(aaaaaa)##+##color(white)(aaaaaaa)##-##color(white)(aaaaaa)##-#

#color(white)(aaaa)##x-x_2##color(white)(aaaaaa)##-##color(white)(aaaaaaa)##-##color(white)(aaaaaa)##+#

#color(white)(aaaa)##f'(x)##color(white)(aaaaaaa)##-##color(white)(aaaaaaa)##+##color(white)(aaaaaa)##-#

#color(white)(aaaa)##f(x)##color(white)(aaaaaaa)##↘##color(white)(aaaaaaa)##↗##color(white)(aaaaaa)##↘#

So,

#f(x)# is convex when #x in ]-oo,0.678]#

and concave when #x in [-0.382,+oo[#