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a) Find p so that the curve y = x 3 + 2 x 2 + p x + 3 has one horizontal tangent line only. b) Find the value of x where this tangent line occurs.

a) Find p so that the curve y = x 3 + 2 x 2 + p x + 3 has one horizontal tangent line only.

b) Find the value of x where this tangent line occurs.

Grade:12

1 Answers

SAGAR SINGH - IIT DELHI
879 Points
10 years ago

Dear nitin,

  • a) The horizontal tangent line has a slope equal to zero. We first need to find the first derivative of y and equate to 0.

    y ' = 3 x 2 + 4 x + p = 0

  • For the above quadratic function to have one solution only, it needs to have its discriminant D equal to zero.

    D = 16 - 12 p = 0

  • Solve the above for p to find

    p = 4 / 3

  • b) To find the value of x where the horizontal tangent line occurs, we need to solve the above quadratic equation 3 x 2 + 4 x + p = 0 for x. It had the discriminant equal to zero, hence

    x = - 4 / 6 = - 2 / 3

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