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

Sue Pemberton Solutions for Chapter: Further Differentiation, Exercise 4: EXERCISE 8B

Author:Sue Pemberton

Sue Pemberton Mathematics Solutions for Exercise - Sue Pemberton Solutions for Chapter: Further Differentiation, Exercise 4: EXERCISE 8B

Attempt the practice questions on Chapter 8: Further Differentiation, Exercise 4: EXERCISE 8B with hints and solutions to strengthen your understanding. Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book solutions are prepared by Experienced Embibe Experts.

Questions from Sue Pemberton Solutions for Chapter: Further Differentiation, Exercise 4: EXERCISE 8B with Hints & Solutions

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

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 values of $a$ and $b = 4 (4) = 16$ .

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.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 13

The curve $y = 2 x^{3} + a x^{2} + b x + 7$ has a stationary point at $(2, - 13)$ . 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 13

The curve $y = 2 x^{3} + a x^{2} + b x + 7$ has a stationary point at $(2, - 13)$ . Find the coordinates of the second stationary point on the curve.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 13

The curve $y = 2 x^{3} + a x^{2} + b x + 7$ has a stationary point at $(2, - 13)$ . Determine the nature of the two stationary points.

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 1 Course Book>Chapter 8 - Further Differentiation>EXERCISE 8B>Q 13

The curve $y = 2 x^{3} + a x^{2} + b x + 7$ has a stationary point at $(2, - 13)$ . Find the coordinates of the point on the curve where the gradient is minimum and state the value of the minimum gradient.

Learn and Prepare for any exam you want !

|

|

|

|

|

|

|

|