Question

Let, for x ER the maximum value of 8x² -8x+7 4x² - 4x+3 be k. The value of [k] is (where [.] denotes greatest integer function)​

Answer 1

The value of [k] is 2.

Let , for x ∈ R , the maximum value of (8x² - 8x + 7)/(4x² - 4x + 3) be k.

we have to find the value of [k] , where [.] denotes greatest integer function.

here, f(x) = (8x² - 8x + 7)/(4x² - 4x + 3)

differentiating with respect to x,

f'(x) = [(4x² - 4x + 3) × d(8x² - 8x + 7)/dx - (8x² - 8x + 7) × d(4x² - 4x + 3)]/(4x² - 4x + 3)²

= [(4x² - 4x + 3) × (16x - 8) - (8x² - 8x + 7)(8x - 4)]/(4x² - 4x + 3)²

= [(64x³ - 64x² + 48x - 32x² + 32x - 24) - (64x³ - 64x² - 56x - 32x² + 32x - 28 )]/(4x² - 4x + 3)²

= (-8x + 4)/(4x² - 4x + 3)²

at f'(x) = 0

⇒(-8x + 4) = 0

⇒x = 1/2

again differentiating with respect to x,

f"(x) = [(4x² - 4x + 3)² × -8 - (-8x + 4) × 2(4x² - 4x + 3)(8x - 4)]/(4x² - 4x + 3)⁴

at x = 1/2 , f"(x) < 0

hence, at x = 1/2 , value of f(x) will be maximum.

∴ maximum value of f(x) = f(1/2)

= (8x² - 8x + 7)/(4x² - 4x + 3)

= (8 × 1/4 - 8 × 1/2 + 7)/(4 × 1/4 - 4 × 1/2 + 3)

= (2 - 4 + 7)/(1 - 2 + 3)

= 5/2

= 2.5

so, k = 2.5

Therefore the value of [k] = [2.5] = 2.