1. Tardigrade
  2. Question
  3. Mathematics
  4. The function f ( x )=2 x 3+α x 2+β x +γ, where α, β, γ ∈ R has local minimum at P ( log 3 t 2, f ( log 3 t 2)) and local maximum at Q ( log 3 t , f ( log 3 t )). If R ((5/2), f ((5/2))) is the point of inflection, then ' t ' is equal to

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Q. The function $f ( x )=2 x ^3+\alpha x ^2+\beta x +\gamma$, where $\alpha, \beta, \gamma \in R$ has local minimum at $P \left(\log _3 t ^2, f \left(\log _3 t ^2\right)\right)$ and local maximum at $Q \left(\log _3 t , f \left(\log _3 t \right)\right)$. If $R \left(\frac{5}{2}, f \left(\frac{5}{2}\right)\right)$ is the point of inflection, then ' $t$ ' is equal to

Application of Derivatives

Solution:

We have $f ^{\prime}( x )=6 x ^2+2 \alpha x +\beta$
$ \Rightarrow f ^{\prime}\left(\log _3 t ^2\right)=0$ and $f ^{\prime}\left(\log _3 t \right)=0$
Now $\log _3 t ^2+\log _3 t =-\frac{2 \alpha}{6} $ (sum of the roots)
$\Rightarrow 3 \log _3 t =\frac{-\alpha}{3}$
Also, $f ^{\prime \prime}( x )=12 x +2 \alpha$
$ f ^{\prime \prime}\left(\frac{5}{2}\right)=0 \Rightarrow 12\left(\frac{5}{2}\right)+2 \alpha=0 \Rightarrow \alpha=-15 $
$\therefore 3 \log _3 t =\frac{15}{3} \Rightarrow t =3^{5 / 3}$