Question

The minimum value of (x - 1)(x - 2)(x – 3) (X - 4) is m then m + 4 is​

Answer 1

The minimum value of (x - 1)(x - 2)(x - 3)(x - 4) is m

we have to find the value of (m + 4).

solution : let y = (x - 1)(x - 2)(x - 3)(x - 4)

= x⁴ - (1 + 2 + 3 + 4)x³ + (1 × 2 + 2 × 3 + 3 × 4 + 4 × 1 + 4 × 2 + 3 × 1)x² - (1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 1 + 4 × 1 × 2)x + 1 × 2 × 3 × 4

= x⁴ - 10x³ + 35x² - 50x +24

now y = x⁴ - 10x³ + 35x² - 50x + 24

differentiating y with respect to x,

dy/dx = 4x³ - 30x² + 70x - 50

= 2(2x³ - 15x² + 35x - 25)

at dy/dx = 0, 2x³ - 15x² + 35x - 25 = 0

⇒x = 5/2, (5 + √5)/2 and (5 - √5)/2

now again differentiating w.r.t to x we get,

d²y/dx² = 12x² - 60x + 70

at x = 5/2 , d²y/dx² < 0 [ get maximum ]

at x = (5 + √5)/2 , d²y/dx² >0 [ get minimum]

at x = (5 - √5)/2, d²y/dx² > 0 [ get minimum]

after evaluating at x = (5 ± √5)/2, we get y = -1

Therefore the minimum value of y = -1 = m

now value of (m + 4) = -1 + 4 = 3

Therefore the value of (m + 4) = 3

Answer 2

Given:-

• The minimum value of (x - 1)(x - 2)(x – 3) (X - 4) is m .

To find:-

• Value of  m + 4 is​ ?

Solution:-

Assume  y = (x - 1)(x - 2)(x - 3)(x - 4)

=> y =  x⁴ - (1 + 2 + 3 + 4)x³ + (1 × 2 + 2 × 3 + 3 × 4 + 4 × 1 + 4 × 2 + 3 × 1)x² - (1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 1 + 4 × 1 × 2)x + 1 × 2 × 3 × 4

=>   y = x⁴ - 10x³ + 35x² - 50x +24

Differentiating ' y ' wrt ' x' ,

=>  dy/dx = 4x³ - 30x² + 70x - 50

        = 2(2x³ - 15x² + 35x - 25)

When  dy/dx = 0,

=>  2x³ - 15x² + 35x - 25 = 0

=> x = 5/2,

(5 + √5)/2 and (5 - √5)/2

again differentiating w.r.t to x ,

=> d²y/dx² = 12x² - 60x + 70

Note :-

• d²y/dx² < 0 we get maximum .

• d²y/dx² > 0  we get minimum.

=> at x = 5/2 , d²y/dx² < 0

at x = (5 + √5)/2 , d²y/dx² < 0

at x = (5 - √5)/2, d²y/dx² > 0

At x = (5 ± √5)/2, we get y = -1

=> Hence,  the minimum value of y = -1 = m

Now,

=>  (m + 4) = -1 + 4 = 3

Hence, the value of (m + 4) = 3

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