Question

The function f(x) = ax² – bx – 4x³ + g, where a, b, g, Ī R, has local maxima at P(log₂a, f(log₂a)) & Local minima at Q (log₂a², f(log ₂a²)). If the graph of f(x) changes concavity about the point R $\left( \frac { 3 } { 4 } , \mathrm { f } \left( \frac { 3 } { 4 } \right) \right)$ , then which of the following conic section can have eccentricity 'a'

• A. Circle
• B. Parabola
• C. Ellipse
• D. ) Hyperbola

Answer 1

)

Sol. f '(x) = 2ax – 12x² – b

log ₂a + 2log ₂a = $\frac { \alpha } { 6 }$

(maxima & minima are roots of f '(x) = 0

log₂a = $\frac { \alpha } { 18 }$ ..........(i)

f "(x) = 2a – 24x = 0 Ž x = $\frac { \alpha } { 12 }$

Ž $\frac { 3 } { 4 }$ = $\frac { \alpha } { 12 }$ Ž a = 9

Ž log ₂a = $\frac { 9 } { 18 }$ (from (i))

\ a = $\sqrt { 2 }$