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  4. If y=x3-ax2+48x+7 is an increasing function for all real values of x, then a lies in:

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Q. If $ y={{x}^{3}}-a{{x}^{2}}+48x+7 $ is an increasing function for all real values of $ x, $ then a lies in:

KEAMKEAM 2000

Solution:

$ \frac{dy}{dx}=3{{x}^{2}}-2ax+48>0 $ ( $ \because $ y is an increasing function) $ \therefore $ Discriminant, $ D<0\,\,4{{a}^{2}}-4\times 3\times 48<0 $ $ \Rightarrow $ $ {{a}^{2}}-144<0\Rightarrow a\in (-12,12) $