Question

A small town is analyzing the pattern of a new street light installation. The lights are set up in such a way that the intensity of light at any point x metres from the start of the street can be modelled by f(x) = ex sin x, where x is in metres.

Based on the above, answer the following:

1. Find the intervals on which the f(x) is increasing or decreasing, x ∈ [0, π]?   [2]
2. Verify, whether each critical point when x ∈ [0, π] is a point of local maximum or local minimum or a point of inflexion.   [2]

Answer 1

(i) f(x) = e^x sin x

f'(x) = e^x cos x + e^x sin x

⇒ e^x (cos x + sin x)

f'(x) = 0

⇒ e^x (cos x + sin x) = 0

e^x ≠ 0

cos x = −sin x

x = `(3pi)/4  "in" [0, pi]`

∴ Inverval `[0, (3pi)/4), ((3pi)/4, pi]`

f' = `((2pi)/3)`

= `e^(2pi//3)(cos  (2pi)/3 + sin  (2pi)/3)   ...[∵ x = (2pi)/3]`

= `e^(2pi//3)((-1)/2 + sqrt3/2) > 0`

Hence, f(x) is increasing on `[0, (3pi)/4)` and decreasing on `((3pi)/4, pi]`

(ii) f"(x) = e^x (cos x + sin x) + e^x (−sin x + cos x)

= 2e^x cos x

`f"''"((3pi)/4) = 2e^(3pi//4) xx (-1)/sqrt2`

= `(-2e^(3pi//4))/sqrt2 < 0`

Hence, the point x = `(3pi)/4` is a local maximum.