Question

What is the complete set of numbers for which f(x) = x3 + 2x2 - x is concave?

Answer 1

Concave up at
`(-(2)/(3),\infty)`

Concave down at
`(-\infty,-(2)/(3))`

Explanation:

The given function is
`f(x)=x^3+2x^2-x`

In order to check the concavity, we find the inflection point. To find inflection point we calculate the second derivative of the function and then equate it to zero.

The first derivative is given by

`f'(x)=3x^2+4x-1`

The second derivative is given by

`f''(x)=6x+4`

For inflection point

`f''(x)=0\\\\6x+4=0\\\\x=-(4)/(6)\\\\x=-(2)/(3)`

Hence, the inflection point is
`x=-(2)/(3)`

Hence, on the left of
`x=-(2)/(3)` the curve is concave downward and on the right of
`x=-(2)/(3)` the curve is concave up.

Hence,

Concave up at
`(-(2)/(3),\infty)`

Concave down at
`(-\infty,-(2)/(3))`

Answer 2

Get the second derivative, and we'll see that the function will be concave up for all positive values of the second derivative and concave down for all negative values. The second derivative we’ll set equal to zero and find out where the inflection point is.

6x+4=0

6x=−4

x=−4/6=−2/3

For all x less than or to the left of -2/3, the function is concave down. For all x greater than -2/3, . the function is concave up to the right of x=−2/3