Q74 / 160 · TS EAMCET 2017 · Mathematics

If the slope of the tangent of the curve $y=a x^3+b x+4$ at $(2,14)$ is 21 , then the values of $a$ and $b$ respectively

  • Correct answer
    A
    $2,-3$
  • B
    $3,-2$
  • C
    $-3,-2$
  • D
    2, 3

Solution

The curve $y=a x^3+b x+4$ is passes through (2, 14). $$ \begin{array}{rlrlrl} & \therefore & & 14 & =a(2)^3+b(2)+4 \\ \Rightarrow & & 14 & =8 a+2 b+4 \\ \Rightarrow & & 5 & =4 a+b \end{array} $$ Slope of tangent to the curve $y=a x^3+b x+4$ i.e. $\frac{d y}{d x}=3 a x^2+b$ $$ \begin{array}{rlrl} \Rightarrow & \left(\frac{d y}{d x}\right)_{(2,14)} & =3 a(2)^2+b \\ \Rightarrow & & 21 & =12 a+b \end{array} $$ $$ \left[\because \frac{d y}{d x}=21\right] $$ Solving Eqs. (i) and (ii), we get $$ a=2, b=-3 $$

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

Relevant Questions