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Answer :
Final answer:
To find the value of df⁻¹/dx at x=69 for the function f(x)=7x³−13x²−3, we differentiate f(x) to get f'(x) and apply the inverse function theorem. After differentiating and simplifying, we find that df⁻¹/dx at x=69 is 1/111.
Explanation:
The question requires us to find the value of df⁻¹/dx at the point x=69=f(3) when given the function f(x)=7x³−13x²−3, with the domain x≥1.5. To find this, we need to apply the concept of the inverse function theorem, which states that if f is differentiable at a point a and f'(a) ≠ 0, then f⁻¹ is differentiable at f(a) and df⁻¹/dx can be found using the formula:
df⁻¹/dx|x=f(a) = 1 / f'(a)
First, we differentiate f(x) with respect to x to find f'(x), which yields:
f'(x) = 21x² - 26x
Then, we evaluate f'(3):
f'(3) = 21(3)² - 26(3) = 189 - 78 = 111.
Finally, we apply the inverse function theorem:
df⁻¹/dx|x=69 = 1 / 111
Therefore, the value of df⁻¹/dx at the point x=69 is 1/111.
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