The curve y = a x + b x 2 has a stationary point at ( 2 , 12 ) . Find the values of a and b .

Given that, the equation of the curve: y = a x + b x 2 Now, differentiate y with respect to x , then we get: d y d x = d d x a x + b x 2 ⇒ d y d x = a d d x x + b d d x 1 x 2 ⇒ d y d x = a - 2 b x 3 For stationary points, we have: d y d x = 0 Therefore, a - 2 b 2 3 = 0 [Since, the curve has stationary point at x = 2 ] ⇒ 8 a - 2 b = 0 ⇒ b = 4 a → 1 Now, at x = 2 , then we have: y = 12 i.e., 12 = a 2 + b 2 2 ⇒ 48 = 8 a + b ⇒ 48 = 8 a + 4 a [Using equation 1 ] ⇒ a = 4 Substitute the value of a = 4 in equation 1 , then we get: b = 4 4 = 16 Hence, the required values of a = 4 , b = 16 .

The curve y = x 2 + a x + b has a stationary point at ( 3 , 5 ) . Find the values of a and b .

Given that, the equation of the curve: y = x 2 + a x + b Now, differentiate y with respect to x , then we get: d y d x = d d x x 2 + a x + b ⇒ d y d x = d d x x 2 + a d d x 1 x + d d x b ⇒ d y d x = 2 x - a x 2 For stationary points, we have: d y d x = 0 Therefore, 2 3 - a 3 2 = 0 [Since, the curve has stationary point at x = 3 ] ⇒ 6 - a 9 = 0 ⇒ a = 54 Now, at x = 3 , then we have: y = 5 i.e., 5 = 9 + a 3 + b ⇒ 5 = 9 + 54 3 + b [Using a = 54 ] ⇒ 5 = 9 + 18 + b ⇒ b = - 22 Hence, the required values of a = 54 , b = - 22 .

E M B I B E

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 10

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AS and A Level

IMPORTANT

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The curve $y = x^{3} + a x^{2} + b$ has a stationary point at $(4, - 27)$ . Determine the nature of the stationary point $(4, - 27)$ .

Important Questions on Further Differentiation

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 10

The curve $y = x^{3} + a x^{2} + b$ has a stationary point at $(4, - 27)$ . Find the coordinates of the other stationary point on the curve and determine the nature of this stationary point.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 10

The curve $y = x^{3} + a x^{2} + b$ has a stationary point at $(4, - 27)$ . Find the coordinates of the point on the curve where the gradient is minimum and state the value of the minimum gradient.

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AS and A Level

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Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 11

The curve $y = a x + \frac{b}{x^{2}}$ has a stationary point at $(2, 12)$ . Find the values of $a$ and $b$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 11

The curve $y = a x + \frac{b}{x^{2}}$ has a stationary point at $(2, 12)$ . Determine the nature of the stationary point $(2, 12)$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 11

The curve $y = a x + \frac{b}{x^{2}}$ has a stationary point at $(2, 12)$ . Find the range of values of $x$ for which $a x + \frac{b}{x^{2}}$ is increasing.

HARD

AS and A Level

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Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 12

The curve $y = x^{2} + \frac{a}{x} + b$ has a stationary point at $(3, 5)$ . Find the values of $a$ and $b$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 12

The curve $y = x^{2} + \frac{a}{x} + b$ has a stationary point at $(3, 5)$ . Determine the nature of the stationary point $(3, 5)$ .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 12

The curve $y = x^{2} + \frac{a}{x} + b$ has a stationary point at $(3, 5)$ . Find the range of values of $x$ for which $x^{2} + \frac{a}{x} + b$ is decreasing.