SATHEE: Application of Derivatives 1 Question 18

Application of Derivatives 1 Question 18

####18. The curve $y = a x^{3} + b x^{2} + c x + 5$ , touches the $X$ -axis at $P (- 2, 0)$ and cuts the $Y$ -axis at a point $Q$ , where its gradient is 3 . Find $a, b, c$ .

(1994, 5M)

Show Answer

Solution:

Given, $y = a x^{3} + b x^{2} + c x + 5$ touches $X$ -axis at $P (- 2, 0)$ which implies that $X$ -axis is tangent at $(- 2, 0)$ and the curve is also passes through $(- 2, 0)$ .

The curve cuts $Y$ -axis at $(0, 5)$ and gradient at this point is given 3 , therefore at $(0, 5)$ slope of the tangent is 3.

Now,

$\frac{d y}{d x} = 3 a x^{2} + 2 b x + c$

Since, $X$ -axis is tangent at $(- 2, 0)$ .

$\begin{aligned} ∴ \frac{d y}{d x}_{x = - 2} = 0 \\ \Rightarrow 0 = 3 a (- 2)^{2} + 2 b (- 2) + c \\ \Rightarrow 0 = 12 a - 4 b + c \end{aligned}$

Again, slope of tangent at $(0, 5)$ is 3 .

$\begin{aligned} ∴ {| \frac{d y}{d x} |}_{(0, 5)} = 3 \\ \Rightarrow 3 = 3 a (0)^{2} + 2 b (0) + c \\ \Rightarrow 3 = c \end{aligned}$

Since, the curve passes through $(- 2, 0)$ .

$\begin{aligned} 0 & = a (- 2)^{3} + b (- 2)^{2} + c (- 2) + 5 \\ \Rightarrow & 0 & = - 8 a + 4 b - 2 c + 5 \end{aligned}$

From Eqs. (i) and (ii),

$12 a - 4 b = - 3$

From Eqs. (ii) and (iii),

$- 8 a + 4 b = 1$

On adding Eqs. (iv) and (v), we get

$4 a = - 2 \Rightarrow a = - 1 / 2$

On putting $a = - 1 / 2$ in Eq. (iv), we get

$\begin{aligned} 12 (- 1 / 2) - 4 b & = - 3 \\ \Rightarrow & - 6 - 4 b & = - 3 \\ \Rightarrow & - 3 & = 4 b \\ \Rightarrow & b & = - 3 / 4 \\ ∴ & a = - 1 / 2, b & = - 3 / 4 and c = 3 \end{aligned}$

Application of Derivatives 1 Question 18

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Application of Derivatives 1 Question 18

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu