Question

Determine the intervals on which the given function is concave up or down and find points of inflection f(x)= (x^2-6)e^x

Answer 1

need to take the 2nd derivative first
f'(x)=(x^2-6)e^x+2xe^x
f''(x)=(x^2-6)e^x+2xe^x+2e^x+2xe^x
f''(x)=e^x[x^2-6+4x+2]
f''(x)=e^x[x^2+4x-4]

Answer 2

now set f''(x)=0
e^x[x^2+4x-4]=0
e^x is never 0, so we'll concern ourselves with
x^2+4x-4=0
using the quadratic formula, i got
\[x=-2-2\sqrt{2}, -2+2\sqrt{2}\]
this splits the domain into three intervals
\[(-\infty, -2-2\sqrt{2}), (-2-2\sqrt{2}, -2+2\sqrt{2}), (-2+2\sqrt{2}, \infty)\]

Answer 3

can you find the solutions from here?

Answer 4

i just needed the intervals ..and if they were concave up or down i couldn't figure out how to find the points after finding the second derivative thank you