find the open intervals where the function is concave upward or concave downward. find any inflection…

find the open intervals where the function is concave upward or concave downward. find any inflection points.\nf(x) = -x² - 8x + 7\nwhere is the function concave upward and where is it concave downward? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function is concave downward on the interval(s) (-∞, ∞). the function is never concave upward.\n(type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. the function is concave upward on the interval(s). the function is never concave downward.\n(type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nc. the function is concave upward on the interval(s) and concave downward on the interval(s).\n(type your answers in interval notation. use integers or fractions for any numbers in the expressions. use a comma to separate answers as needed.)\nd. the function is never concave upward or downward.\nfind any inflection points of f. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function has an inflection point at.\n(type an ordered pair, using integers or fractions. use a comma to separate answers as needed.)\nb. the function has no inflection points.

find the open intervals where the function is concave upward or concave downward. find any inflection points.\nf(x) = -x² - 8x + 7\nwhere is the function concave upward and where is it concave downward? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function is concave downward on the interval(s) (-∞, ∞). the function is never concave upward.\n(type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. the function is concave upward on the interval(s). the function is never concave downward.\n(type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nc. the function is concave upward on the interval(s) and concave downward on the interval(s).\n(type your answers in interval notation. use integers or fractions for any numbers in the expressions. use a comma to separate answers as needed.)\nd. the function is never concave upward or downward.\nfind any inflection points of f. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function has an inflection point at.\n(type an ordered pair, using integers or fractions. use a comma to separate answers as needed.)\nb. the function has no inflection points.

Answer

Explanation:

Step1: Find the first and second derivatives

Given ( f(x)=-x^{2}-8x + 7). The first - derivative (f^{\prime}(x)) using the power rule ((x^{n})^\prime=nx^{n - 1}): (f^{\prime}(x)=\frac{d}{dx}(-x^{2}-8x + 7)=-2x-8). The second - derivative (f^{\prime\prime}(x)): (f^{\prime\prime}(x)=\frac{d}{dx}(-2x - 8)=-2).

Step2: Analyze concavity

Since (f^{\prime\prime}(x)=-2<0) for all (x\in(-\infty,\infty)). A function (y = f(x)) is concave upward when (f^{\prime\prime}(x)>0) and concave downward when (f^{\prime\prime}(x)<0). So the function (f(x)) is never concave upward ((f^{\prime\prime}(x)) is never positive) and is concave downward on ((-\infty,\infty)).

Step3: Analyze inflection points

Inflection points occur where (f^{\prime\prime}(x)) changes sign. Since (f^{\prime\prime}(x)=-2) (a constant), there is no value of (x) for which (f^{\prime\prime}(x)) changes sign.

Answer:

  • For concavity: B. The function is concave upward on the interval(s) ( ). The function is never concave upward. The function is concave downward on the interval(s) ((-\infty,\infty)).
  • For inflection points: B. The function has no inflection points.