Graph the function and locate intervals on which the function is increasing or decreasing, open intervals on which the function is concave up or down, and all inflection points WITHOUT USING A CALCULATOR. x^4 e^x , -infinity

Question

Graph the function and locate intervals on which the function is increasing or decreasing, open intervals on which the function is concave up or down, and all inflection points WITHOUT USING A CALCULATOR. x^4 e^x , -infinity<x<infinity

anonymous · Answer

$$f(x)=x^4e^{x}$$
$$f\prime(x)=4x^3e^x+x^4e^x=e^x(4x^3+x^4)$$
$$f\prime(x)=12x^2e^x+4x^3e^x+4x^3e^x+x^4e^x=4x^2e^x(3+2x+x^2)$$

now set both derivatives equal to zero,
so 
$$f\prime(x)=0 \;\;\; \rightarrow \;\;\; 0=e^x(4x^3+x^4) \;\;\; \rightarrow \;;\;\ x=\{0, -4\}$$

now do the intervals, pick test points and draw the chart with the signs....I think you can keeep going from here.

anonymous · Answer

the third function is the second derivative
$$f\prime\prime(x)$$
sorry, typo.

anonymous · Answer

How did you calculate the second derivative?

anonymous · Answer

product rule twice on the first derivative.

anonymous · Answer

I'm talking about this part 4x^2e^x(3+2x+x^2)

anonymous · Answer

oh....just group them together and the factor the GCF.