Question

(i) Which of the following function is Increasing for all values of x in its domain?

(a) sin x

(b) log x

(c) x²

(d) |x|

(ii) Find a point on the curve y = (x – 2)² at which the tangent is parallel to the chord joining the points (2,0) and (4,4).

(iii) Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 – 24x – 6x².

Answer 1

(i) (b) log x

(ii) Given; y = (x - 2)² ⇒ \(\frac{dy}{dx}\) = 2(x - 2)

Slope of the chord = \(\frac{4-0}{4-2}\) = 2

⇒ 2 = 2(x - 2) ⇒ x = 3 ⇒ y = (3 - 2)² = 1

Therefore the required point is (3,1)

(iii) Given; p(x) = 41 – 24x – 6x²

p’(x) = – 24 – 12x

p”(x) = – 12

For turning points p’(x) = – 24 – 12x = 0

⇒ x = -2

Since p”(x) = – 12 always maximum Therefore maximum value 

p(- 2) = 41 – 24(- 2) 6(- 2)² = 65