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f(c) is a minimum value of f(x) if -...

f(c) is a minimum value of f(x) if -

A

`f'(c) = 0 , f'' (c) gt 0`

B

`f'(c) = 0 , f'' (c) lt 0`

C

`f ' (c) ne 0 , f''(c) = 0`

D

`f '(c) lt 0 , f ''(c) gt 0`

Text Solution

AI Generated Solution

The correct Answer is:
To determine the conditions under which \( f(c) \) is a minimum value of \( f(x) \), we can follow these steps: ### Step 1: Find the First Derivative To find the critical points of the function \( f(x) \), we need to compute the first derivative \( f'(x) \). ### Step 2: Set the First Derivative to Zero We set \( f'(c) = 0 \) to find the critical points. This indicates that the slope of the tangent to the curve at point \( c \) is zero, suggesting a potential minimum or maximum. ### Step 3: Determine the Nature of the Critical Point To determine whether the critical point \( c \) is a minimum, we need to examine the second derivative \( f''(x) \). ### Step 4: Compute the Second Derivative Calculate \( f''(c) \). ### Step 5: Apply the Second Derivative Test - If \( f''(c) > 0 \), then \( f(c) \) is a local minimum. - If \( f''(c) < 0 \), then \( f(c) \) is a local maximum. - If \( f''(c) = 0 \), the test is inconclusive. ### Conclusion Thus, \( f(c) \) is a minimum value of \( f(x) \) if: 1. \( f'(c) = 0 \) 2. \( f''(c) > 0 \) ### Summary of Conditions The conditions for \( f(c) \) to be a minimum are: - \( f'(c) = 0 \) - \( f''(c) > 0 \)
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