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

The temperature, $T$ degrees Celsius, of a cooling laser component $t$ minutes after activation is modeled by the function $T = f (t)$ for $t > 0$ .

It is given that:

the point $P (4, 10)$ lies on the curve $T = f (t)$
the rate of change of temperature is $12\sqrt{t} + \frac{k}{t^2}$ , where $k$ is a constant
the rate of change of temperature, $f^{'} (t)$ , has a stationary point at $P$

Find the exact value of $k$ .

[3]

Determine an expression for $f (t)$ , giving your answer in its simplest form.

[5]

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

The temperature, $T$ degrees Celsius, of a cooling laser component $t$ minutes after activation is modeled by the function $T = f (t)$ for $t > 0$ .

It is given that:

the point $P (4, 10)$ lies on the curve $T = f (t)$
the rate of change of temperature is $12\sqrt{t} + \frac{k}{t^2}$ , where $k$ is a constant
the rate of change of temperature, $f^{'} (t)$ , has a stationary point at $P$

Find the exact value of $k$ .

[3]

Determine an expression for $f (t)$ , giving your answer in its simplest form.

[5]