how to find the exact value of when the second derivative equals 0

Question

Let f(x) = 9xe^x

(a) Find the exact value of x so that f ''(x) = 0.

so I found the second derivative to be f''(x)=9e^x(x+2).

I plugged in zero and got 18, but it said that was wrong. Is there something in my process that I did wrong here?

b) for which intervals of the function is concave up? (put answer in interval notation)

when I graphed this it showed that the graph approached four but in my table it shows and increase and decrease which confuses me. I was say from [0,infinity) but I got it wrong. please help me?

Answer 1

f"(x) = 9e^x[2+x]  so f"(x) = 0 is true only if x=-2.

So you must "plug in"  -2..

 

Part b...concavity is revealed by behavior of f"

The first factor (before square brackets that I used)is always positive, so the sign behavior depends on where x is in relation to -2.

 

On the interval from -infinity up to -2, f" is negative (therefore concavity of f is down) and to the right of that f has upward concavity.

Answer 2

(a) Yes there is. Plugging in zero finds the value of the second derivative at x = 0. But you want to know where the second derivative evaluates to 0. So instead you want to be setting f''(x) = 0, i.e.

9e^x(x+2) = 0

and solving for x. Dividing both sides by 9e^x, we get x+2 = 0, or x = -2.

 

(b) The function is concave up on an interval if its second derivative is > 0 at every point on that interval. Since the second derivative is continuous, and only crosses 0 at x = -2, we simply test any one point on either side of -2 to see if it is > 0 on the respective intervals (-∞,-2) and (-2,∞). Clearly f''(x)>0 for x > -2, and f''(x)<0 for x < -2, so it is concave up on just the interval (-2,∞). Graphing the function f confirms this.  

Answer 3

(a). If f"(x) = 9e^x(x+2) = 0, then x+2 = 0 (since e^x > 0 for all x)

 

          So, x = -2

 

(b).  When x < -2, f"(x) < 0, so the graph of y = f(x) is concave down on        the interval (-∞,-2).

 

       When x > -2, f"(x) > 0, so the graph of y = f(x) is concave up on            the interval (-2,∞).