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Murrr4er [49]
1 year ago
13

Maths Questions box 51.The function f: R→ R is twice differentiable and for any x ER we have g(x) = f(4-x²), if f'(1) = -5 and f

" (1) = -1 than what is g" (√3)?
​
Mathematics
1 answer:
mrs_skeptik [129]1 year ago
5 0

By the chain rule,

g(x) = f(4 - x^2) \\\\ \implies g'(x) = -2x f'(4 - x^2) \\\\ \implies g''(x) = -2 f'(4 - x^2) + 4x^2 f''(4 - x^2)

Observe that

4 - x^2 = 1 \implies x^2 = 3 \implies x = \pm\sqrt3

Then

g''(\sqrt3) = -2 f'(1) + 4\left(\sqrt3\right)^2 f''(1) = -2(-5) + 4(3)(-1) = \boxed{-2}

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