# Math Help - Concavity

1. ## Concavity

For the function f(x)=(x^(2)+4x+(11/4)e^(x) I need to find where it is concave up.

So, I found the first derivative and then the second derivative which is,
f''(x)= e^(x)(x^(2)+8x+(51/4))
Then, I found the roots of the function using the quadratic formula which is (-8+/-sqrt(13))/(2)

I found it was concave upwards at the open interval: (((-8+sqrt(13))/2), inf)

But somehow this is incorrect. Can someone help me?

2. Originally Posted by ctran
For the function f(x)=(x^(2)+4x+(11/4)e^(x) I need to find where it is concave up.

So, I found the first derivative and then the second derivative which is,
f''(x)= e^(x)(x^(2)+8x+(51/4))
Then, I found the roots of the function using the quadratic formula which is (-8+/-sqrt(13))/(2)

I found it was concave upwards at the open interval: (((-8+sqrt(13))/2), inf)

But somehow this is incorrect. Can someone help me?
is the function ...

$f(x) = \left(x^2 + 4x + \frac{11}{4}\right)e^x
$

or

$f(x) = x^2 + 4x + \frac{11}{4}e^x
$

3. Originally Posted by skeeter
is the function ...

$f(x) = \left(x^2 + 4x + \frac{11}{4}\right)e^x
$

or

$f(x) = x^2 + 4x + \frac{11}{4}e^x
$

Hello skeeter,
The function is this one,

$f(x) = \left(x^2 + 4x + \frac{11}{4}\right)e^x
$

4. Originally Posted by ctran
Hello skeeter,
The function is this one,

$f(x) = \left(x^2 + 4x + \frac{11}{4}\right)e^x
$
your 2nd derivative is incorrect ...

$f''(x) = e^x\left(x^2+6x+\frac{27}{4}\right)$

5. Originally Posted by skeeter
your 2nd derivative is incorrect ...

$f''(x) = e^x\left(x^2+6x+\frac{27}{4}\right)$

No I dont think so, $f''(x) = e^x\left(x^2+6x+\frac{27}{4}\right)$ is actually the first derivative.

I am certain the second derivative is $f''(x) = e^x\left(x^2+8x+\frac{51}{4}\right)$

6. Originally Posted by ctran
No I dont think so, $f''(x) = e^x\left(x^2+6x+\frac{27}{4}\right)$ is actually the first derivative.

I am certain the second derivative is $f''(x) = e^x\left(x^2+8x+\frac{51}{4}\right)$
you're right ... I copied the wrong one down.

$f''(x) = e^x\left(x^2+8x+\frac{51}{4}\right)$

$f''(x) = 0$ at $x = \frac{-8 \pm \sqrt{13}}{2} \approx -2.2$ and $-5.8$

for x < -5.8 , f''(x) > 0 ... concave up

for -5.8 < x < -2.8 , f''(x) < 0 ... concave down

for x > -2.8 , f''(x) > 0 ... concave up