A smooth function is defined and differentiable for all real numbers
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A smooth function g is defined and differentiable for all real numbers. Selected points of its graph are known: g(0)=1 and g(6)=7. Between x=0 and x=6 the graph is increasing, twice continuously differentiable, and exhibits a single gentle maximum near x=4 before descending slightly and then rising again to x=6. A careful sketch shows that the straight line joining the endpoints of the interval appears to lie entirely below the curve from x=0 to approximately x=2, intersects the curve only at the endpoints, and that around x≈2.3 the slope of the curve looks identical to the slope of this straight line.
Answer the following questions about the interval [0,6].
(a) Verify that all hypotheses required to guarantee the existence of a point c in (0,6) whose tangent line has the same slope as the secant line through (0,1) and (6,7) are met for g on [0,6]. Provide a brief justification for each hypothesis.
(b) Determine the exact value of the slope of the secant line through (0,1) and (6,7).
(c) Explain why at least one number c in (0,6) must satisfy g′(c) equal to the value found in part (b). Your answer should rely on the conclusions you reached in part (a).
(d) Using only the verbal description of the graph above, estimate an x-coordinate at which such a point c is likely to occur. Describe the graphical evidence that supports your estimate and discuss how confident you are that another point with the same property does not exist elsewhere on (0,6). No algebraic computation of g′(x) is required.