Sketching Graphs of Functions and Their Derivatives Quiz (MCQs)

Let f be the function defined by f⁡(x)=1/3⁢x^3−4⁢x^2−9⁢x+5. On which of the following intervals is the graph of f both decreasing and concave down?

(-1,4)

The first derivative of the function ℎ is given by ℎ′⁡(x)=3⁢ln⁡(2+cos⁡(2⁢x))−x, and the second derivative of ℎ is given by ℎ″⁡(x)=−6⁢sin⁡(2⁢x)/2+cos⁡(2⁢x)−1. On what open intervals contained in −2<x<2 is the graph of ℎ both increasing and concave down?

(-2,-1.486) and (-0.250,1.085)

Let f be the function defined by f(x)=1/3⁢x^3−3⁢x^2−16⁢x. On which of the following intervals is the graph of f both decreasing and concave down?

(-2,3) only

The first derivative of the function ℎ is given by ℎ′⁡(x)=sin⁡x+cos⁡(x^2)+x, and the second derivative of ℎ is given by ℎ″⁡(x)=cos⁡x−2⁢x⁢sin⁡(x^2)+1. On what open intervals contained in −3<x<2 is the graph of ℎ both increasing and concave down?

(0.969,1.697) only

The function f is continuous on the interval (0,16), and f is twice differentiable except at x=3, where the derivatives are undefined. Information about the first and second derivatives of f for values of x in the interval (0,16) is given in the table above. At what values of x in the interval (0,16) does the graph of f have a point of inflection?

x=3 and x=9

The function f is continuous on the interval (0,16), and f is twice differentiable except at x=5 where the derivatives are undefined. Information about the first and second derivatives of f for values of x in the interval (0,16) is given in the table above. At what values of x in the interval (0,16) does the graph of f have a point of inflection?

x=5 and x=8