Question

What will be the domain of the function, if 3^x + 3^f(x) = minimum of Φ(t), where Φ(t) = minimum of (2t^3 – 15t^2 + 36t – 25, 2 + |sint|; 2 ≤ t ≤ 4} ?

(a) (-∞, 1)

(b) (-∞, loge3)

(c) (0, loge2)

(d) (-∞, loge2)

I have been asked this question in a national level competition.

This question is from Second Order Derivative topic in chapter Limits and Derivatives of Mathematics – Class 11

Answer 1

The correct option is (d) (-∞, loge2)

Easy explanation: Let, g(t) = 2t^3 – 15t^2 + 36t -25

g’(t) = 6t^2 – 30t + 36 = 0

=> 6(t^2 – 5t + 6) = 0

=> t = 2, 3

For, 2 ≤ t ≤ 4, at t = 3, g”(t) > 0

So, g(t) is minimum.

g(t)min = g(3) = 2*27 – 15*9 + 36*3 – 25 = 2

Also, 2 + |sint| ≥ 2

Hence, minimum Φ(t) = 2

Therefore, 3^x + 3^f(x) = 2

=>3^f(x) = 2 – 3^x

Thus, 3^f(x) > 0

=> 3^x > 0 and 3^x < 2

Therefore, x € (-∞, loge2)