MCQ
ધારો કે $f _{\lambda}( x )=4 \lambda x ^{3}-36 \lambda x ^{2}+36 x +48$ એ પ્રત્યેક $x \in R$ માટે વધતું હોય તેવી $\lambda$ ની મહતમ કિમત $\lambda^{*}$ છે .તો $f _{\lambda} *(1)+ f _{\lambda} *(-1)$ = ..........
- A$36$
- B$48$
- C$64$
- D$72$
$f_{\lambda}^{\prime}(x)=12 \lambda x^{2}-72 \lambda x+36$
$f_{\lambda}^{\prime}(x)=12\left(\lambda x^{2}-6 \lambda x+3\right) \geq 0$
$\therefore \lambda>0 \ and \,D \leq 0$
$36 \lambda^{2}-4 \times \lambda \times 3 \leq 0$
$9 \lambda^{2}-3 \lambda \leq 0$
$3 \lambda(3 \lambda-1) \leq 0$
$\lambda \in\left[0, \frac{1}{3}\right]$
$\therefore \lambda_{\text {largest }}=\frac{1}{3}$
$f ( x )=\frac{4}{3} x ^{3}-12 x ^{2}+36 x +48$
$\therefore f (1)+ f (1)=72$
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