14 real exam questions with step-by-step video solutions · Try each question yourself first, then open the video solution to check your working. Stuck? Ask Axi below – it knows this page’s questions.
New · Axi
Stuck on a stationary points question?
Axi, our AI tutor, answers questions on any stationary point problem on this page – no copying, no pasting, no sign-up.
What are you stuck on? Ask about any stationary points question on this page and I will talk you through it.
Axi answers from this page’s question, tutorial and worked solution. Live Axi reads and marks your own working in the app.
1Edexcel A Level June 2018 Paper 1 Q2
A curve C has equation y = x^2 - 2x - 24sqrt(x), x > 0. (a) Find (i) dy/dx (ii) the second derivative d^2y/dx^2. (3) (b) Verify that C has a stationary point when x = 4. (2) (c) Determine the nature of this stationary point, giving a reason for your answer. (2)
View video solution (1 part)
Part (a):
2Edexcel A Level Specimen Paper 1 Q1
The curve C has equation y = 3x^4 - 8x^3 - 3. (a) Find (i) dy/dx (ii) the second derivative d^2y/dx^2. (3) (b) Verify that C has a stationary point when x = 2. (2) (c) Determine the nature of this stationary point, giving a reason for your answer. (2)
View video solution (4 parts)
Part (a):
Part (b):
Part (c):
Part (d):
3CIE P1 June 2013 Q9
A curve has equation y = f(x) and is such that f'(x) = 3x^(1/2) + 3x^(-1/2) - 10. (i) By using the substitution u = x^(1/2), or otherwise, find the values of x for which the curve y = f(x) has stationary points. [4] (ii) Find f''(x) and hence, or otherwise, determine the nature of each stationary point. [3] (iii) It is given that the curve y = f(x) passes through the point (4, -7). Find f(x). [4]
View video solution (3 parts)
Part (a):
Part (b):
Part (c):
4EDEXCEL C2 January 2013 Q8
The curve C has equation y = 6 - 3x - 4/x^3, x not equal to 0. (a) Use calculus to show that the curve has a turning point P when x = sqrt(2). (4) (b) Find the x-coordinate of the other turning point Q on the curve. (1) (c) Find d^2y/dx^2. (1) (d) Hence or otherwise, state with justification, the nature of each of these turning points P and Q. (3)
View video solution (1 part)
Part (a):
5OCR C1 June 2012 Q8
(i) Find the coordinates of the stationary point on the curve y = x^4 + 32x. [5] (ii) Determine whether this stationary point is a maximum or a minimum. [2] (iii) For what values of x does x^4 + 32x increase as x increases? [1]
View video solution (1 part)
Part (a):
6OCR C1 January 2012 Q7
A curve has equation y = (x + 2)(x^2 - 3x + 5). (i) Find the coordinates of the minimum point, justifying that it is a minimum. [8] (ii) Calculate the discriminant of x^2 - 3x + 5. [2] (iii) Explain why (x + 2)(x^2 - 3x + 5) is always positive for x > -2. [2]
View video solution (2 parts)
Part (a):
Part (b):
7EDEXCEL C2 January 2010 Q9
The curve C has equation y = 12sqrt(x) - x^(3/2) - 10, x > 0. (a) Use calculus to find the coordinates of the turning point on C. (7) (b) Find d^2y/dx^2. (2) (c) State the nature of the turning point. (1)
View video solution (3 parts)
Part (a):
Part (b):
Part (c):
8EDEXCEL C2 June 2008 Q8
Figure 2 shows a sketch of part of the curve with equation y = 10 + 8x + x^2 - x^3. The curve has a maximum turning point A. (a) Using calculus, show that the x-coordinate of A is 2. (3) The region R, shown shaded in Figure 2, is bounded by the curve, the y-axis and the line from O to A, where O is the origin. (b) Using calculus, find the exact area of R. (8)
View video solution (2 parts)
Part (a):
Part (b):
9Edexcel A Level June 2022 Paper 2 Q6
Figure 2 shows a sketch of part of the curve with equation y = f(x) where f(x) = 8sin(x/2) - 3x + 9, x > 0 and x is measured in radians. The point P, shown in Figure 2, is a local maximum point on the curve. Using calculus and the sketch in Figure 2, (a) find the x coordinate of P, giving your answer to 3 significant figures. (4) The curve crosses the x-axis at x = alpha, as shown in Figure 2. Given that, to 3 decimal places, f(4) = 4.274 and f(5) = -1.212, (b) explain why alpha must lie in the interval [4, 5]. (1) (c) Taking x_0 = 5 as a first approximation to alpha, apply the Newton-Raphson method once to f(x) to obtain a second approximation to alpha. Show your method and give your answer to 3 significant figures. (2)
View video solution (1 part)
Part (a):
10Edexcel A Level June 2022 Paper 1 Q15
A company makes toys for children. Figure 5 shows the design for a solid toy that looks like a piece of cheese. The toy is modelled so that face ABC is a sector of a circle with radius r cm and centre A; angle BAC = 0.8 radians; faces ABC and DEF are congruent; edges AD, CF and BE are perpendicular to faces ABC and DEF; edges AD, CF and BE have length h cm. Given that the volume of the toy is 240 cm^3, (a) show that the surface area of the toy, S cm^2, is given by S = 0.8r^2 + 1680/r, making your method clear. (4) Using algebraic differentiation, (b) find the value of r for which S has a stationary point. (4) (c) Prove, by further differentiation, that this value of r gives the minimum surface area of the toy. (2)
View video solution (1 part)
Part (a):
11Edexcel A Level June 2019 Paper 2 Q11
Figure 8 shows a sketch of the curve C with equation y = x^x, x > 0. (a) Find, by firstly taking logarithms, the x coordinate of the turning point of C. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5) The point P(alpha, 2) lies on C. (b) Show that 1.5 < alpha < 1.6. (2) A possible iteration formula that could be used in an attempt to find alpha is x_(n+1) = 2x_n^(1 - x_n). Using this formula with x_1 = 1.5, (c) find x_4 to 3 decimal places, (2) (d) describe the long-term behaviour of x_n. (2)
View video solution (1 part)
Part (a):
12Edexcel A Level June 2023 Paper 1 Q4
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The curve C has equation y = f(x) where x is real. Given that f'(x) = 2x + (1/2)cos(x), the curve has a stationary point with x coordinate alpha, and alpha is small, (a) use the small angle approximation for cos(x) to estimate the value of alpha to 3 decimal places. (3) The point P(0, 3) lies on C. (b) Find the equation of the tangent to the curve at P, giving your answer in the form y = mx + c, where m and c are constants to be found. (2)
13Edexcel A Level June 2023 Paper 2 Q5
The curve C has equation y = f(x). The curve passes through the point P(3, -10) and has a turning point at P. Given that dy/dx = 2x^3 - 9x^2 + 5x + k where k is a constant, (a) show that k = 12. (2) (b) Hence find the coordinates of the point where C crosses the y-axis. (3)
14Edexcel AS June 2023 Paper 1 Q16
A curve has equation y = f(x), x >= 0. Given that f'(x) = 4x + a*sqrt(x) + b, where a and b are constants, the curve has a stationary point at (4, 3), and the curve meets the y-axis at -5, find f(x), giving your answer in simplest form. (6)
