Question

The temperature of a person during an intestinal illness is given by f(x)=−0.1x²+mx+98.60,≤x≤12,m being a constant, where f(x) is the temperature in ∘F at x days. Is the function differentiable in the interval (0, 12)?

Answer 1

The function
`\( f(x) = -0.1x^2 + mx + 98.60 \)` is differentiable in the interval (0, 12) for any constant m. The differentiability is not affected by the specific value of m in this range.

To determine the differentiability of the function
`\( f(x) = -0.1x^2 + mx + 98.60 \)` in the interval (0, 12), we need to check the differentiability criteria.

The function
`\( f(x) \)` is differentiable at a point x = c if the limit

`\[ \lim_{{h \to 0}} (f(c + h) - f(c))/(h) \]`

exists. If this limit exists for all x in the interval (0, 12), then f(x) is differentiable in that interval.

Let's find the derivative of f(x):

`\[ f'(x) = -0.2x + m \]`

The derivative is a linear function, and it is defined for all values of x.

Therefore, f(x) is differentiable in the interval (0, 12) for any value of the constant m. The differentiability is not dependent on the value of m in this case.