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Question
If f(x) = x5 + 2x – 3, then (f–1)1 (–3) = ______.
Options
0
– 3
`-1/3`
`1/2`
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Solution
If f(x) = x5 + 2x – 3, then (f–1)1 (–3) = `bbunderline(1/2)`.
Explanation:
Given:
f(x) = x5 + 2x − 3
We need to find (f−1)′(−3)
Step 1: Formula for derivative of inverse function
(f−1)′(a) = `1/(f'(f^(-1)(a)))`
So we must find the value of x such that f(x) = a = −3.
Step 2: Find x such that f(x) = −3
x5 + 2x − 3 = −3
Simplify:
x5 + 2x = 0
x(x4 + 2) = 0
So x = 0.
Step 3: Compute f′(x)
f′(x) = 5x4 + 2
At x = 0,
f′(0) = 5(0)4 + 2 = 2
Step 4: Substitute in the formula
(f−1)′(−3) = `1/(f'(0)) = 1/2`
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