Question

The curve y = 4x² + 2x − 8 and y = x³ − x + 13 touch each other at the point _________________.

Answer 1

Disclaimer: The solution has been provided for the following question.

The curve y = 4x² + 2x − 8 and y = x³ − x + 10 touch each other at the point _________________.


Solution:

Let the given curves each other at the point (h, k).

∴ k = 4h² + 2h − 8 .....(1)

k = h³ − h + 10 .....(2)

Consider the first curve C₁ ≡ y = 4x² + 2x − 8.

y = 4x² + 2x − 8

$\Rightarrow \frac{dy}{dx}=8x+2$

∴ Slope of tangent to the curve C₁ at (h, k) = ${\left(\frac{dy}{dx}\right)}_{\left(h,k\right)}=8h+2$

Consider the second curve C₂ ≡ y = x³ − x + 10.

y = x³ − x + 10

$\Rightarrow \frac{dy}{dx}=3x^2-1$

∴ Slope of tangent to the curve C₂ at (h, k) = ${\left(\frac{dy}{dx}\right)}_{\left(h,k\right)}=3h^2-1$

It is given that, the two curves touch each other at (h, k).

∴ Slope of tangent to the curve C₁ at (h, k) = Slope of tangent to the curve C₂ at (h, k)

$\Rightarrow 8h+2=3h^2-1$

$\Rightarrow 3h^2-8h-3=0$

$\Rightarrow 3h^2-9h+h-3=0$

$\Rightarrow 3h\left(h-3\right)+1\left(h-3\right)=0$

$\Rightarrow \left(3h+1\right)\left(h-3\right)=0$

⇒ 3h + 1 = 0 or h − 3 = 0

⇒ h = $-\frac{1}{3}$ or h = 3

Putting h = $-\frac{1}{3}$ in (1), we get

$k=4{\left(-\frac{1}{3}\right)}^2+2\times \left(-\frac{1}{3}\right)-8=\frac{4}{9}-\frac{2}{3}-8=-\frac{74}{9}$

Putting h = $-\frac{1}{3}$ in (2), we get

$k={\left(-\frac{1}{3}\right)}^3-\left(-\frac{1}{3}\right)+10=-\frac{1}{27}+\frac{1}{3}+10=\frac{278}{27}$

Here, the values of k are not same for the two curves for h = $-\frac{1}{3}$.

So, the two curves do not touch each other when h = $-\frac{1}{3}$. However, the tangents are parallel to each other at h = $-\frac{1}{3}$.

Putting h = 3 in (1), we get

$k=4\times {\left(3\right)}^2+2\times 3-8=36+6-8=34$

Putting h = 3 in (2), we get

$k={\left(3\right)}^3-3+10=27-3+10=34$

Here, the values of k are same for the two curves when h = 3.

So, the two curves touch each other at (3, 34).

Thus, the curves y = 4x² + 2x − 8 and y = x³ − x + 10 touch each other at the point (3, 34).


The curve y = 4x² + 2x − 8 and y = x³ − x + 10 touch each other at the point ___(3, 34)___.