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1.9.17 Concavity and the second derivative (A-level only)

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Question 30

The height of a biological culture, H(t)H(t)H(t) millimetres, is observed over time t t\,t hours for t>0t > 0t>0.

Given that

  • H′(t)=10t14+kt2−4\displaystyle H'(t) = 10t^{\frac{1}{4}} + \frac{k}{t^2} - 4H′(t)=10t41​+t2k​−4, where k k\,k is a constant
  • H′′(t)=0H''(t) = 0H′′(t)=0 when t=16t = 16t=16
a.

Determine the value of kkk.

[3]
b.

Given also that

  • H(t)=20H(t) = 20H(t)=20 when t=1t = 1t=1

Find an expression for H(t)H(t)H(t), giving each term in simplest form.

[4]

1.9.17 Concavity and the second derivative (A-level only) Questions

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