Find the intervals of concavity for f(x) = x^(4)e^(−x).

Question

Also identify any inflection points (p, f(p)).?

Answer 1

The Function is .

Differentiate with respect to x.

Again Differentiate with respect to x.

Find the x-values at which f''(x) = 0

As e^(-x) cannot be equated to zero then

x²(x²-8x+12) = 0

then x² = 0 and x²-8x+12 = 0

if x² = 0 then roots of x are 0,0

if x²-8x+12 = 0 then the roots are

x²-6x -2x+12 = 0

x(x-6)-2(x-6) = 0

(x-6)(x-2) = 0.

x = 6,2.

if x²-8x+12 = 0 then the Roots are 6,2

Therefore Inflection points are are 0, 0, 2 and 6.

Therefore the test intervals are assumed as (-∞ , 0) , (0 , 2) , (2,6) and (6, ∞)

Interval    Test Value                          Sign of f''(x)                                                    Conclusion

(-∞ , 0)    x = -1          Concave upward

(0,2)        x = 1                      Concave upward

(2,6)        x = 3             Concave downward

(6,∞ )      x = 7                  Concave upward.