KVPY Sample Paper KVPY Stream-SX Model Paper-17

Let \[f(x)=a{{x}^{3}}+b{{x}^{2}}+cx+d,\,a>0,a,b,c,d\in R\] and \[f(x)=0\] has all roots of repeated nature. If \[g(x)=f'(x)-f''(x)+f'''(x)\] then \[\forall \,x\in R\]

A) \[g(x)>0\]

B) \[g(x)\ge 0\]

C) \[g(x)<0\]

D) \[g(x)\le 0\]

Correct Answer: A

Solution :

Clearly the equation \[f'(x)=0\] must also have repeated roots. So, \[f'(x)\ge 0\,\forall \,x.\]

Let \[f'(x)={{a}_{1}}{{x}^{2}}+{{b}_{1}}x+{{c}_{1}},\]

Where \[{{a}_{1}}>0\] and \[b_{1}^{2}-4{{a}_{1}}{{c}_{1}}=0,\] then \[g(x)={{a}_{1}}{{x}_{2}}+({{b}_{1}}-2{{a}_{1}})x+2{{a}_{1}}-{{b}_{1}}+{{c}_{1}}\]

Its discriminant \[b_{1}^{2}-4{{a}_{1}}{{c}_{1}}-4a_{1}^{2}<0\] \[\Rightarrow \]\[g(x)>0\,\forall \,x\in R\]

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KVPY Sample Paper KVPY Stream-SX Model Paper-17

question_answer Let \[f(x)=a{{x}^{3}}+b{{x}^{2}}+cx+d,\,a>0,a,b,c,d\in R\] and \[f(x)=0\] has all roots of repeated nature. If \[g(x)=f'(x)-f''(x)+f'''(x)\] then \[\forall \,x\in R\] A) \[g(x)>0\] B) \[g(x)\ge 0\] C) \[g(x)<0\] D) \[g(x)\le 0\]

Let \[f(x)=a{{x}^{3}}+b{{x}^{2}}+cx+d,\,a>0,a,b,c,d\in R\] and \[f(x)=0\] has all roots of repeated nature. If \[g(x)=f'(x)-f''(x)+f'''(x)\] then \[\forall \,x\in R\]

A) \[g(x)>0\]

B) \[g(x)\ge 0\]

C) \[g(x)<0\]

D) \[g(x)\le 0\]