The length x of a rectangle is decreasing at the rate of 5 cm/minute, and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rate of change of (a) the perimeter, (b) the area of the rectangle.

OR

Find the intervals in which the function f Given: by f(x) = sin x + cos x, 0 ≤ × ≤ 2π, is strictly increasing or strictly decreasing.

Given: , and x = 8 cm and y = 6 cm

(a) Perimeter of rectangle

We know that perimeter of rectangle = 2 (l + b)

= 2 (x + y)

Differentiating both sides w. r. to t, we get

= 2 (-5 + 4)

= -2 cm/min

∴ Perimeter of rectangle is decreasing at the rate of 2 cm/min.

(b) Area of rectangle

We know that area of rectangle = lb

= xy

Differentiating both sides w. r. to t, we get

= 8 (4) + 6 (-5)

= 32 – 30

= 2 cm²/min

∴ Area of rectangle is increasing at the rate of 2 cm²/min.

OR

Given: f (x) = sin x + cos x, 0 ≤ x ≤ 2π

∴ f’ (x) = cos x – sin x

We know that for stationary points, f’ (x) = 0

∴ cos x – sin x = 0

⇒ tan x =1

∴ x = π/4, 5π/4 as 0 ≤ x ≤ 2π

The points x = π/4 and x = 5π/4 divide the interval [0, 2π] into following disjoint intervals: (0, π/4), (π/4, 5π/4) and (5π/4, 2π)

Here f is strictly increasing in the intervals (0, π/4) and (5π/4, 2π) and f is strictly decreasing in the interval (π/4, 5π/4).