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

Question

The function f ( x )=2 x 3+α x 2+β x +&gamma;, where α, β, &gamma; ∈ 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 (A) 32 / 5 (B) 31 / 5 (C) 35 / 3 (D) 33 / 5

Tardigrade Admin · Accepted Answer

We have f prime( x )=6 x 2+2 α x +β ⇒ f prime( log 3 t 2)=0 and f prime( log 3 t )=0 Now log 3 t 2+ log 3 t =-(2 α/6) (sum of the roots) ⇒ 3 log 3 t =(-α/3) Also, f prime prime( x )=12 x +2 α f prime prime((5/2))=0 ⇒ 12((5/2))+2 α=0 ⇒ α=-15 ∴ 3 log 3 t =(15/3) ⇒ t =35 / 3

Q. The function $f (x) = 2 x^{3} + α x^{2} + β x + γ$ , where $α, β, γ \in R$ has local minimum at $P (lo g_{3} t^{2}, f (lo g_{3} t^{2}))$ and local maximum at $Q (lo g_{3} t, f (lo g_{3} t))$ . If $R (\frac{5}{2}, f (\frac{5}{2}))$ is the point of inflection, then ' $t$ ' is equal to

$3^{2/5}$

$3^{1/5}$

$3^{5/3}$

$3^{3/5}$

Q. The function f(x)=2x3+αx2+βx+γ, where α,β,γ∈R has local minimum at P(log3​t2,f(log3​t2)) and local maximum at Q(log3​t,f(log3​t)). If R(25​,f(25​)) is the point of inflection, then ' t ' is equal to

32/5

31/5

35/3

33/5

We have f′(x)=6x2+2αx+β ⇒f′(log3​t2)=0 and f′(log3​t)=0 Now log3​t2+log3​t=−62α​ (sum of the roots) ⇒3log3​t=3−α​ Also, f′′(x)=12x+2α f′′(25​)=0⇒12(25​)+2α=0⇒α=−15 ∴3log3​t=315​⇒t=35/3

Q. The function $f (x) = 2 x^{3} + α x^{2} + β x + γ$ , where $α, β, γ \in R$ has local minimum at $P (lo g_{3} t^{2}, f (lo g_{3} t^{2}))$ and local maximum at $Q (lo g_{3} t, f (lo g_{3} t))$ . If $R (\frac{5}{2}, f (\frac{5}{2}))$ is the point of inflection, then ' $t$ ' is equal to

$3^{2/5}$

$3^{1/5}$

$3^{5/3}$

$3^{3/5}$