Consider the polynomial function f(x)=3 x4-4 x3-12 x2+5. This function has monotonicity as given below: in (-∞, a1) decreasing in (a1, a2) increasing in (a2, a3) decreasing in (a3, ∞) increasing A rectangle A B C D is formed such that ℓ(A B)= portion of the tangent to the curve y=f(x) at x=a1, intercepted between the lines x=a1 x = a 3. ℓ( BC )= portion of the line x = a 3 intercepted between the curve x-axis. Triplet ( a 1, a 2, a 3) is given by -

Question

Consider the polynomial function f(x)=3 x4-4 x3-12 x2+5. This function has monotonicity as given below: in (-∞, a1) decreasing in (a1, a2) increasing in (a2, a3) decreasing in (a3, ∞) increasing A rectangle A B C D is formed such that ℓ(A B)= portion of the tangent to the curve y=f(x) at x=a1, intercepted between the lines x=a1 x = a 3. ℓ( BC )= portion of the line x = a 3 intercepted between the curve x-axis. Triplet ( a 1, a 2, a 3) is given by - (A) (-1,0,2) (B) (0,-1,2) (C) (2,-1,0) (D) (2,0,-1)

Tardigrade Admin · Accepted Answer

f(x)= 3 x4-4 x3-12 x2+5 f prime(x) =12 x3-12 x2-24 x =12 x(x-2)(x+1) ∴ a1=-1, a2=0 a3=2 . <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/e53a75f8dca09674c99eba5a07581c44-.png" /> on the basis of above graph, the given questions can be solved.

$(- 1, 0, 2)$

$(0, - 1, 2)$

$(2, - 1, 0)$

$(2, 0, - 1)$

$f (x) = 3 x^{4} - 4 x^{3} - 12 x^{2} + 5$
$f^{'} (x) = 12 x^{3} - 12 x^{2} - 24 x$
$= 12 x (x - 2) (x + 1)$
$∴ a_{1} = - 1, a_{2} = 0& a_{3} = 2.$

on the basis of above graph, the given questions can be solved.

(−1,0,2)

(0,−1,2)

(2,−1,0)

(2,0,−1)

f(x)=3x4−4x3−12x2+5 f′(x)=12x3−12x2−24x =12x(x−2)(x+1) ∴a1​=−1,a2​=0&a3​=2. on the basis of above graph, the given questions can be solved.

$(- 1, 0, 2)$

$(0, - 1, 2)$

$(2, - 1, 0)$

$(2, 0, - 1)$

$f (x) = 3 x^{4} - 4 x^{3} - 12 x^{2} + 5$
$f^{'} (x) = 12 x^{3} - 12 x^{2} - 24 x$
$= 12 x (x - 2) (x + 1)$
$∴ a_{1} = - 1, a_{2} = 0& a_{3} = 2.$

on the basis of above graph, the given questions can be solved.