If p(x) =ax²+bx+c has no real zeroes and 16a+ 4b+ c < 0 then:  (a) c=0  (b) c>0 (c) c<0  (d) none of these

Dear Student,

Please find below the solution to the asked query :
$$\begin{gathered} \mathrm{p}\left(\mathrm{x}\right)= {\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c} \\ \mathrm{p}\left(4\right)= 16\mathrm{a}+4\mathrm{b}+\mathrm{c} \\ \mathrm{Now} \mathrm{p}\left(4\right)=16\mathrm{a}+4\mathrm{b}+\mathrm{c}<0 \\ \mathrm{p}\left(4\right)<0 \\ \mathrm{and} \mathrm{it} \mathrm{has} \mathrm{no} \mathrm{real} \mathrm{zeroes} \mathrm{so} \mathrm{it} \mathrm{will} \mathrm{be} \mathrm{either} \mathrm{above} \mathrm{x}-\mathrm{axis} \mathrm{or} \mathrm{below} \mathrm{x}-\mathrm{axis} \\ \mathrm{but} \mathrm{since} \mathrm{p}\left(4\right)<0 , \mathrm{it} \mathrm{is} \mathrm{definetely} \mathrm{below} \mathrm{x}-\mathrm{axis} \mathrm{so} \mathrm{c}<0 \end{gathered}$$


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