f: [-5,5]→[3,18] Surjective & differentiable

To solve the problem, we need to analyze the function \( f \) and its derivatives based on the given conditions. Let's break it down step by step.

Understanding the Function

The function \( f: [-5, 5] \rightarrow [3, 18] \) is surjective, meaning it covers the entire range from 3 to 18. It is also differentiable, which implies that its first derivative \( f' \) exists everywhere in the interval.

Key Conditions

• We need to find the least integral value of \( k \) such that there exists a point \( c \) where certain conditions hold.
• The conditions involve the first and second derivatives of \( f \) at point \( c \).

Analyzing the Derivative Conditions

The conditions given are:

• \((f'(c))^2 = -f(c) f''(c)\)
• \((f(c))^2 f''(c) < 3(f'(c))^3\)

From the first condition, we can infer that \( f(c) \) must be negative or zero for \( f''(c) \) to be positive, which is not possible since \( f(c) \) ranges from 3 to 18. Thus, we need to ensure that \( f'(c) \) is zero at some point to satisfy the equation.

Finding the Value of k

To satisfy the second condition, we need to analyze the behavior of \( f(c) \) and its derivatives. Since \( f(c) \) is always positive, we can rewrite the second condition as:

\( f''(c) < \frac{3(f'(c))^3}{(f(c))^2} \)

To find the least integral value of \( k \), we can set \( g(0) = k \) and analyze the behavior of \( f \) around \( c = 0 \). By ensuring that \( f(0) \) is within the range of 3 to 18, we can find suitable values for \( k \).

Conclusion

After evaluating the conditions and ensuring that \( f \) meets the requirements, the least integral value of \( k \) that satisfies all conditions is:

k = 3