# IB Math Analysis & Approaches Question bank-Topic: SL 5.8 Points of inflexion with zero and non-zero gradients SL Paper 1

## Question

The diagram shows part of the graph of $$y = f'(x)$$ . The x-intercepts are at points A and C. There is a minimum at B, and a maximum at D. (i)     Write down the value of $$f'(x)$$ at C.

(ii)    Hence, show that C corresponds to a minimum on the graph of f , i.e. it has the same x-coordinate.


a(i) and (ii).

Which of the points A, B, D corresponds to a maximum on the graph of f ?


b.

Show that B corresponds to a point of inflexion on the graph of f .


c.

## Markscheme

(i) $$f'(x) = 0$$     A1     N1

(ii) METHOD 1

$$f'(x) < 0$$ to the left of C, $$f'(x) > 0$$ to the right of C     R1R1     N2

METHOD 2

$$f”(x) > 0$$     R2     N2

[3 marks]

a(i) and (ii).

A     A1     N1

[1 mark]

b.

METHOD 1

$$f”(x) = 0$$     R2

discussion of sign change of $$f”(x)$$     R1

e.g. $$f”(x) < 0$$ to the left of B and $$f”(x) > 0$$ to the right of B; $$f”(x)$$ changes sign either side of B

B is a point of inflexion     AG     N0

METHOD 2

B is a minimum on the graph of the derivative $${f’}$$     R2

discussion of sign change of $$f”(x)$$     R1

e.g. $$f”(x) < 0$$ to the left of B and $$f”(x) > 0$$ to the right of B; $$f”(x)$$ changes sign either side of B

B is a point of inflexion     AG     N0

[3 marks]

c.

## Question

A function f has its first derivative given by $$f'(x) = {(x – 3)^3}$$ .

Find the second derivative.


a.

Find $$f'(3)$$ and $$f”(3)$$ .


b.

The point P on the graph of f has x-coordinate $$3$$. Explain why P is not a point of inflexion.


c.

## Markscheme

METHOD 1

$$f”(x) = 3{(x – 3)^2}$$     A2     N2

METHOD 2

attempt to expand $${(x – 3)^3}$$     (M1)

e.g. $$f'(x) = {x^3} – 9{x^2} + 27x – 27$$

$$f”(x) = 3{x^2} – 18x + 27$$     A1     N2

[2 marks]

a.

$$f'(3) = 0$$ , $$f”(3) = 0$$     A1     N1

[1 mark]

b.

METHOD 1

$${f”}$$ does not change sign at P     R1

evidence for this     R1     N0

METHOD 2

$${f’}$$ changes sign at P so P is a maximum/minimum (i.e. not inflexion)     R1

evidence for this     R1     N0

METHOD 3

finding $$f(x) = \frac{1}{4}{(x – 3)^4} + c$$ and sketching this function     R1

indicating minimum at $$x = 3$$     R1     N0

[2 marks]

c.