Question:
The absolute maximum value of \(f(x) = x^2 + 1\) in \([-5, 2]\) is
The absolute maximum value of \(f(x) = x^2 + 1\) in \([-5, 2]\) is
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Compare f at the critical point x = 0 and at both end points x = -5 and x = 2. The largest value is the absolute maximum.
Updated On: Oct 22, 2026
- 26
- 1
- 5
- 2
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
The absolute maximum of a continuous function on a closed interval is the largest value among the values at the critical points inside the interval and the values at the two end points.
Step 2: Find the critical points.
The function is \(f(x) = x^2 + 1\).
Differentiate it: \(f^{\prime}(x) = 2x\).
Set \(f^{\prime}(x) = 0\). This gives \(x = 0\), which lies inside \([-5, 2]\).
Step 3: Evaluate f at the critical point and the end points.
\[ f(0) = 0^2 + 1 = 1 \]
\[ f(-5) = (-5)^2 + 1 = 26 \]
\[ f(2) = 2^2 + 1 = 5 \]
The values are 1, 26 and 5. The largest is 26.
Step 4: Why the other options are wrong.
Option (B), 1, is the value at \(x = 0\). It is the absolute minimum, not the maximum.
Option (C), 5, is the value at the right end point \(x = 2\). It is smaller than 26.
Option (D), 2, is the right end point itself, not a value of the function.
Final Answer:
The absolute maximum value is 26, at \(x = -5\). This is option (A). \[ \boxed{26} \]
The absolute maximum of a continuous function on a closed interval is the largest value among the values at the critical points inside the interval and the values at the two end points.
Step 2: Find the critical points.
The function is \(f(x) = x^2 + 1\).
Differentiate it: \(f^{\prime}(x) = 2x\).
Set \(f^{\prime}(x) = 0\). This gives \(x = 0\), which lies inside \([-5, 2]\).
Step 3: Evaluate f at the critical point and the end points.
\[ f(0) = 0^2 + 1 = 1 \]
\[ f(-5) = (-5)^2 + 1 = 26 \]
\[ f(2) = 2^2 + 1 = 5 \]
The values are 1, 26 and 5. The largest is 26.
Step 4: Why the other options are wrong.
Option (B), 1, is the value at \(x = 0\). It is the absolute minimum, not the maximum.
Option (C), 5, is the value at the right end point \(x = 2\). It is smaller than 26.
Option (D), 2, is the right end point itself, not a value of the function.
Final Answer:
The absolute maximum value is 26, at \(x = -5\). This is option (A). \[ \boxed{26} \]
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