Q65 / 160 · AP EAMCET 2014 · Mathematics

The condition that $f(x)=a x^3+b x^2+c x+d$ has no extreme value, is

  • A
    $b^2>3 a c$
  • B
    $b^2=4 a c$
  • C
    $b^2=3 a c$
  • Correct answer
    D
    $b^2 < 3 a c$

Solution

Given curve is $$ f(x)=a x^3+b x^2+c x+d $$ On differentiating w.r.t. $x$, we get $$ f^{\prime}(x)=3 a x^2+2 b x+c $$ For extremum, $f^{\prime}(x)=0$ $$ \therefore \quad 3 a x^2+2 b x+c=0 $$ Since, it has no extremum value $$ \begin{array}{cl} \therefore & b^2-4 a c < 0 \\ \Rightarrow & (2 b)^2-4 \times 3 a \times c < 0 \\ \Rightarrow & 4 b^2-12 a c < 0 \\ \Rightarrow & b^2-3 a c < 0 \\ \Rightarrow & b^2 < 3 a c \end{array} $$

Hence, the correct answer is option: 'D'.

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