# Answer to Question #8383 in Calculus for Kay

Question #8383

Find the intervals on which the graph of f is concave upward, the intervals on which the graph of f is concave downward and the inflection points.

f(x)= ln (x^2-8x+52)

f(x)= ln (x^2-8x+52)

Expert's answer

f'(x) = (2x-8)/(x²-8x+52);

f"(x) = 2/(x²-8x+52)-(2x-8)²/(x²-8x+52)² = -2(x²-8x-20)/(x²-8x+52)²

(x²-8x+52)²>0 (x²-8x+52≠0 cause discriminant is negative) for any x;

x²-8x-20 = 0 ==> x = 10, x = -2 - inflection points;

f"(x) > 0, x<-2 ==> f(x) is concave upward,

f"(x) < 0, -2<x<10 ==> f(x) is concave downward,

f"(x) > 0, 10<x ==> f(x) is concave upward.

f"(x) = 2/(x²-8x+52)-(2x-8)²/(x²-8x+52)² = -2(x²-8x-20)/(x²-8x+52)²

(x²-8x+52)²>0 (x²-8x+52≠0 cause discriminant is negative) for any x;

x²-8x-20 = 0 ==> x = 10, x = -2 - inflection points;

f"(x) > 0, x<-2 ==> f(x) is concave upward,

f"(x) < 0, -2<x<10 ==> f(x) is concave downward,

f"(x) > 0, 10<x ==> f(x) is concave upward.

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