AP EAMCET · Maths · Application of Derivatives

If the curves $y = x^{3} - 3 x^{2} - 8 x - 4$ and $y = 3 x^{2} + 7 x + 4$ touch each other at a point $P$ then the equation of common tangent at $P$ is

A $x - y + 1 = 0$
B $2 x - y + 1 = 0$
C $x + y + 1 = 0$
D $2 x + y + 1 = 0$

Verified Solution

Answer & Solution

Correct Answer

(A) $x - y + 1 = 0$

Step-by-step Solution

Detailed explanation

Equating the two given curveto find of intersection we have, 3x2+7x+4=x3-3x2-8x-4 ⇒x3-6x2-15x-8=0 ⇒x+12x-8=0 Now putting x=-1 in the y=3x2+7x+4 equation, we get y=3-7+4=0, Now putting x=8 in the equation y=3x2+7x+4 we get y=252 So, the points are 8,252 and -1,0 Now…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If the curves y=x3-3x2-8x-4 and y=3x2+7x+4 touch each other at a point P then the equation of common tangent at P is

Step-by-step Solution

If the curves $y = x^{3} - 3 x^{2} - 8 x - 4$ and $y = 3 x^{2} + 7 x + 4$ touch each other at a point $P$ then the equation of common tangent at $P$ is