Determine whether the function f(x)=x−1x+52 is concave up or concave down and its intervals?

Question

Declan Cameron · Accepted Answer

Step 1Using calculus, the general method of determining concavity is to investigate the sign of the second derivative.f(x)=x−1x+52f′(x)=53(x+52)2f″(x)=−106(x+52)3For this function, the sign of f&#039;&#039; is the opposite of the sign of x+52f″ is positive on the interval (−∞, −52) and negative on (−52, ∞)So the graph of f is concave up interval (−∞, −52) and concave down on (−52, ∞)Because -52 is not in the domain of f, there is no inflection point.(The definition of inflection point that I am accustomed to is: a point on the graph at which the concavity changes.)f(x)=x−1x+52 can be written:f(x)=(x+52)−53x+52=x+52x+52−53x+52=1−53x+52From the graph of y=1x

Determine whether the function \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}-{1}}}{{{x}+{52}}}}\) is concave

Answered question

Answer & Explanation

New Questions in Calculus 1