The intensity of light (in lux) measured along a straight rod emanating from a small…
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
The intensity I(x) of light (in lux) measured along a straight rod emanating from a small lamp is modeled by
I(x)=(x+1)e−x,0≤x≤6,
where x is the distance in meters from the lamp.
(a) Verify that I is continuous on [0,6] and differentiable on (0,6).
(b) Determine all critical points of I on (0,6) and show that exactly one such point exists.
(c) Compute I′′ at the critical point found in part (b) and classify the nature of the extremum at that point.
(d) Evaluate I at the interval endpoints and use your results together with part (c) to decide whether the critical point gives an absolute maximum or an absolute minimum on [0,6]. Justify your answer.
(e) State the exact distance from the lamp at which the absolute extremum occurs and its corresponding intensity value. Explain in words what this result means for an observer moving along the rod.