A twice–differentiable function is defined on the open interval
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A twice–differentiable function g is defined on the open interval (−2,4).
On that interval it is known that
• g′(x)>0 for −2<x<−1 and 0<x<1,
• g′(x)=0 at x=−1 and x=1,
• g′(x)<0 for −1<x<0 and 1<x<4,
• g′′(x)>0 for −2<x<0 and 2<x<4,
• g′′(x)<0 for 0<x<2.
No other information about g is given.
Answer the following questions about the graph of y=g(x) on (−2,4).
(1) Determine the intervals on which g is increasing and the intervals on which g is decreasing. Provide justification based on the given information.
(2) State the x–coordinates of all relative extrema of g in (−2,4) and classify each as a local maximum or a local minimum. Explain your reasoning.
(3) Determine the intervals on which the graph of g is concave up and the intervals on which it is concave down. Identify all inflection points and justify their locations.
(4) Using only the information above, describe qualitatively how the graph of g behaves on (−2,4). Your description should indicate the turning and bending behavior but does not require specific numerical values.