Question
The function f(x)=x3 β62x2+ax+9
has a local maximum at x=1. Then the value of a is:Solution
Given the function f(x)=x3 β62x2+ax+9. If f(x) has a local maximum at x=1, then the first derivative fβ²(x) must be zero at x=1, and the second derivative fβ²β²(x) must be negative at x=1. First, find the first derivative of f(x): Since there is a local maximum at x=1, we must have fβ²(1)=0: fβ²(1) = 3(1)2 β124(1) + a = 0 3 β 124 + a = 0 β121 + a = 0 a = 121 Now, we need to check the second derivative to ensure it's a local maximum. Find the second derivative of f(x):
Evaluate the second derivative at x=1: fββ(1) = 6(1)β124 fββ(1) = 6β124 fββ(1) = β118 Since fββ(1) = β118 < 0, there is indeed a local maximum at x=1. Therefore the final answer is option (B), a = 121.
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