Question

\(2f\left( {\frac{x}{2}} \right) + 3f\left( {\frac{{2 - x}}{3}} \right) = g(x)\), \(0 \leqslant x < 3\), \(f''(x) > 0;\) if g(x) is strictly increasing in (a, b) and g(x) is strictly decreasing in (c, d) then 25 ad – bc is
1. 14
2. 15
3. 16
4. 19

Answer 1

Correct Answer - Option 2 : 15

\(g'(x) = f'\left( {\frac{x}{2}} \right) - f'\left( {\frac{{2 - x}}{3}} \right) \Rightarrow f''(x) > 0 \Rightarrow f'(x) \uparrow \)

\(g'(x) > 0 \Rightarrow \frac{x}{2} > \frac{{2 - x}}{3} \Rightarrow x > \frac{4}{5}\)

\(f\left( x \right) \uparrow \left( {\frac{4}{5},3} \right),f\left( x \right) \downarrow \left( {0,\frac{4}{5}} \right)\)