Consider the function below. (If an answer does not exist, enter DNE.)

Question

Consider the function below. (If an answer does not exist, enter DNE.)

C(x) = x^1/5(x + 6)

Find the inflection points.

smaller x value and larger x value

Find the intervals where the graph is concave upward. (Enter your answer using interval notation.)

 

Find the interval where the graph is concave downward. (Enter your answer using interval notation.)

Answer 1

Domain of this function is all reaal numbers, or (- ∞, ∞) So smoller and larger x-value DNE.

f(x) = x^6/5 + 6x^1/5; f'(x) = 6/5x^1/5 + 6/5x^-4/5;

^{f''(x) = 6/25x^-4/5 - 24/25x^(-9/5) = 0; x = 4 or x = 0}

^{f''(x)= 6/25(x - 4)/x^(9/5)}

^{f''(x) > 0 for intervals (- ∞, 0), (4, ∞) graph f(x) concave up}

^{f''(x) < 0 for interval (0, 4) graph f(x) concave down}

^{(0,0) and (4, 10·4^(1/5)) are inflection points.}