Find the intervals on which the graph of f is concave...

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) = In (x\textsuperscript{2} - 6x + 25) For what interval(s) of x is the graph of f concave upward? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (-1,7) (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as needed.) B. The graph is never concave upward. For what interval(s) of x is the graph of f concave downward? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as needed.) B. The graph is never concave downward.

Transcript

00:01 Suppose f of x equals the natural log of x squared minus ata x plus 52.

00:05 For this problem we are to find the intervals where this function is concave upward, where it is concave downward, and the inflection points of this function.

00:18 So to do that, we first want to take the derivatives of this function until the second derivative.

00:25 Now, f prime of x by the chain rule is equal to 1 over x squared minus 8x plus 52 times 2x minus 8th.

00:39 So since this is equal to 2x minus 8 over x squared minus 8x plus 52, then to get the second derivative, we have to apply the quotient rule that will give us x squared minus 8x plus 52, 8x plus 52 times derivative of 2x minus 8 which is 2 minus 2 minus 2x minus 8 times the derivative of x squared minus 8x plus 52 which is 2x minus 8 all over the square of x squared minus 8x plus 52 this is equal to 2x squared minus 16x plus 104 minus 4x squared plus 32x minus 64 and then we divide this by the square of x squared minus 8x plus 52 can simplify that into negative 2x squared plus 16x plus 40 all over the square of x squared minus 8x plus 50 and then we want to set the second derivative to 0 and solve for x.

02:05 We have negative 2x squared plus 16x plus 40, all over the square of x squared minus 8x plus 52 equals 0.

02:18 We will get negative 2x squared plus 16x plus 40 equals 0.

02:26 Factor out negative 2, that's negative 2, times 2, x squared minus 8x minus 20 equals 0 that's negative 2 times x minus 10 times x plus 2 equals 0 so we get x equals 10 and x equals negative 2 now we will use these x values to partition the real number line given that the domain of f here is all real numbers using x equals negative 2 and 10.

03:02 Say this is our number line.

03:07 You have negative infinity, infinity, then you have negative 2 and then 10.

03:12 So the intervals are from negative infinity to negative 2, negative 2 to 10, and then 10 to positive infinity.

03:19 And then you pick test values from each interval.

03:21 Say negative 3 here, zero here, and then 11 here.

03:24 And then we find the sign of f double prime.

03:28 So the denominator is already positive due to the power 2.

03:36 And since this is the same as negative 2 times x minus 10 times x plus 2 over the square of x squared minus 8x plus 52, then this is equal to a number having the sign negative over positive value here.

03:59 So if x, let's say x is negative 3, then x minus n is negative, and then x plus 2 is also negative...

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