​ (a) The slope of the tangent to the curve given x = 1−cosθ, y = θ−sinθ by at θ = π/2

Question

(a) The slope of the tangent to the curve given

x = 1−cosθ, y = θ−sinθ by at θ = \(\frac{π}{2}\)

(i) 0

(ii) – 1

(iii) 1

(iv) Not defined.

(b) Find the intervals in which the function f(x) = x² – 4x + 6 is strictly decreasing.

(C) Find the minimum and maximum value, if any, of the function f(x) = (2x – 1)² + 3

Answer 1

(a) (iii) 1

(b) Given; f(x) = x² – 4x + 6 ⇒ f’(x) = 2x – 4

For turning points; f’(x) = 2x – 4 0 ⇒ x = 2

So volurn.,e is niaxirnum when h = 2r

The intervals are (- ∞, 2); (2, ∞)

f’(0) = 2 x 0 – 4 = -4

Therefore f(x) is decreasing in (- ∞, 2)

(c) f(x) = (2 x 1)² + 3

⇒ f’(x) 2(2x – 1) x 2 f”(x) = 8

For tuming points; f’(x) = 8x – 4 = 0 ⇒ x = 1/2

f(x) has minimum value at x = 1/2 minimum value is

f(\(\frac{1}{2}\)) = 3