Let be a twice-differentiable function whose first derivative is defined for
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let h be a twice-differentiable function whose first derivative h′ is defined for −8≤x≤9. A detailed verbal description of the graph of h′ is given below. Unless stated otherwise, h′ is continuous and differentiable everywhere on (−8,9).
• h′(x)=0 at x=−6,−2,4,8 and only at these x-values.
• On (−8,−6), h′(x)>0 and h′ is increasing.
• On (−6,−2), h′(x)<0 and h′ is decreasing.
• On (−2,1), h′(x)>0 and h′ is decreasing.
• On (1,4), h′(x)>0 and h′ is increasing.
• On (4,8), h′(x)<0 and h′ is increasing.
• On (8,9), h′(x)>0 and h′ is decreasing.
In addition, h(0)=3.
(1) Create a complete sign chart for h′ on the interval [−8,9]. Clearly indicate the sign of h′ on each subinterval determined by the critical points listed above.
(2) Based on your sign chart, determine all intervals where h is increasing and all intervals where h is decreasing. Provide interval notation and justify each interval by referencing your sign chart.
(3) Identify the x–coordinates of all relative extrema of h and classify each as a relative maximum or a relative minimum. Justify your classifications using the First Derivative Test.
(4) Sketch a possible graph of h on [−8,9] that is consistent with all given information, including the condition h(0)=3. Label the relative extrema on your sketch. (A precise scale is not required, but qualitative features must be accurate.)