1. Tardigrade
  2. Question
  3. Mathematics
  4. 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 -

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Q. Consider the polynomial function
$f(x)=3 x^4-4 x^3-12 x^2+5$. This function has monotonicity as given below :
in $\left(-\infty, a_1\right)$ decreasing
in $\left(a_1, a_2\right)$ increasing
in $\left(a_2, a_3\right)$ decreasing
in $\left(a_3, \infty\right)$ increasing
A rectangle $A B C D$ is formed such that
$\ell(A B)=$ portion of the tangent to the curve $y=f(x)$ at $x=a_1$, intercepted between the lines $x=a_1$ $\& x = a _3$.
$\ell( BC )=$ portion of the line $x = a _3$ intercepted between the curve & $x$-axis.
Triplet $\left( a _1, a _2, a _3\right)$ is given by -

Application of Derivatives

Solution:

$f(x)= 3 x^4-4 x^3-12 x^2+5 $
$f^{\prime}(x) =12 x^3-12 x^2-24 x $
$=12 x(x-2)(x+1) $
$ \therefore a_1=-1, a_2=0 \& a_3=2 . $
image
on the basis of above graph, the given questions can be solved.