Question:
Let \(f(x)\) be the differentiable function for all \(x\) such that \(f^'(x)\leq 5\) and \(f(1) = 4\). The maximum value of \(f(5)\) is...
Let \(f(x)\) be the differentiable function for all \(x\) such that \(f^'(x)\leq 5\) and \(f(1) = 4\). The maximum value of \(f(5)\) is...
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Apply the Mean Value Theorem on \([1,5]\).
Updated On: Oct 1, 2026
- \(16\)
- \(20\)
- \(24\)
- \(25\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
The Mean Value Theorem says there is a \(c\in(1,5)\) with \(f(5) - f(1) = f'(c)(5-1)\). The function is differentiable, so the theorem applies.
Step 2: Bound:
\(f'(c)\le5\), so \(f(5) - 4 = 4f'(c) \le 20\), giving \(f(5)\le24\).
The maximum value \(24\) is reached by \(f(x) = 5x - 1\), where \(f'(x) = 5\) and \(f(1) = 4\).
Final Answer:
The maximum value of \(f(5)\) is \(24\), option (C). \[ \boxed{24} \]
The Mean Value Theorem says there is a \(c\in(1,5)\) with \(f(5) - f(1) = f'(c)(5-1)\). The function is differentiable, so the theorem applies.
Step 2: Bound:
\(f'(c)\le5\), so \(f(5) - 4 = 4f'(c) \le 20\), giving \(f(5)\le24\).
The maximum value \(24\) is reached by \(f(x) = 5x - 1\), where \(f'(x) = 5\) and \(f(1) = 4\).
Final Answer:
The maximum value of \(f(5)\) is \(24\), option (C). \[ \boxed{24} \]
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