Question

If m is the minimum value of k for which the function f(x) = x√(kx - x²) is increasing in the interval [0,3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to : (A) (5, 3 6) (B) (4, 3 2)
(C) (3, 3 3) (D) (4, 3 3)

Answer 1

When k = m, then the ordered pair (m, M) is equal to :

(D) (4, 3√3)

This can be calculated as follows:

• As per the question,                f(x) = x√(kx−x²)​

• Differentiating both the sides, we get,

        f′(x) = ​(3kx−4x²) ​ /  2​√(kx−x²)

• Now, f′(x)  ≥  0 3kx−4x²  ≥  0

        kx−x²  ≥  0 4x²−3kx  ≤  0

        x²−kx  ≤  0 4x(x−3k / 4​)  ≤  0

        x(x−k)  ≤  0 so x∈[0,3] 3−3k/4​  ≤  0

• Thus, we get,

        +ve x  ≥  3 k  ≥  4

• Therefore, minimum value of k is-

        m = 4

• f(x) = x√(kx − x²)

                   = 3√(4x3 - 3²)

                   ​= 3√3

• Therefore, maximum value of k is-

        M = 3√3

• Hence, (m,M) = (4, 3√3)