Date | November 2020 | Marks available | 7 | Reference code | 20N.2.SL.TZ0.S_10 |
Level | Standard Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Find | Question number | S_10 | Adapted from | N/A |
Question
Consider a function f(x), for x≥0. The derivative of f is given by f'(x)=6xx2+4.
The graph of f is concave-down when x>n.
Show that f''(x)=24-6x2(x2+4)2.
Find the least value of n.
Find ∫6xx2+4dx.
Let R be the region enclosed by the graph of f, the x-axis and the lines x=1 and x=3. The area of R is 19.6, correct to three significant figures.
Find f(x).
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
METHOD 1
evidence of choosing the quotient rule (M1)
eg vu'-uv'v2
derivative of 6x is 6 (must be seen in rule) (A1)
derivative of x2+4 is 2x (must be seen in rule) (A1)
correct substitution into the quotient rule A1
eg 6(x2+4)-(6x)(2x)(x2+4)2
f''(x)=24-6x2(x2+4)2 AG N0
METHOD 2
evidence of choosing the product rule (M1)
eg vu'+uv'
derivative of 6x is 6 (must be seen in rule) (A1)
derivative of (x2+4)-1 is -2x(x2+4)-2 (must be seen in rule) (A1)
correct substitution into the product rule A1
eg 6(x2+4)-1+(-1)(6x)(2x)(x2+4)-2
f''(x)=24-6x2(x2+4)2 AG N0
[4 marks]
METHOD 1 (2nd derivative) (M1)
valid approach
eg f''<0,
(exact) A1 N2
METHOD 2 (1st derivative)
valid attempt to find local maximum on (M1)
eg sketch with max indicated,
(exact) A1 N2
[2 marks]
evidence of valid approach using substitution or inspection (M1)
eg
A2 N3
[3 marks]
recognizing that area (seen anywhere) (M1)
recognizing that their answer to (c) is their (accept absence of ) (M1)
eg
correct value for (seen anywhere) (A1)
eg
correct integration for (seen anywhere) (A1)
adding their integrated expressions and equating to (do not accept an expression which involves an integral) (M1)
eg
(A1)
A1 N4
[7 marks]