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Dmitry [639]
3 years ago
9

Please help me with this question

Mathematics
1 answer:
likoan [24]3 years ago
6 0

Answer:

  P = (2, 3)

Step-by-step explanation:

P is a local minimum, so can be found from the upper zero of the derivative.

  y' = 0 = 3x^2 -8x +4

  0 = (3x -2)(x -2)

  x = 2/3, x = 2

So, the x-coordinate of P is 2. The y-coordinate is found by evaluating the function:

  y = ((x -4)x +4)x +3 = ((2 -4)2 +4)2 +3 = 3

The coordinates of P are (2, 3).

__

A graphing calculator confirms this result.

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By first principles, the derivative is

\displaystyle\lim_{h\to0}\frac{(x+h)^n-x^n}h

Use the binomial theorem to expand the numerator:

(x+h)^n=\displaystyle\sum_{i=0}^n\binom nix^{n-i}h^i=\binom n0x^n+\binom n1x^{n-1}h+\cdots+\binom nnh^n

(x+h)^n=x^n+nx^{n-1}h+\dfrac{n(n-1)}2x^{n-2}h^2+\cdots+nxh^{n-1}+h^n

where

\dbinom nk=\dfrac{n!}{k!(n-k)!}

The first term is eliminated, and the limit is

\displaystyle\lim_{h\to0}\frac{nx^{n-1}h+\dfrac{n(n-1)}2x^{n-2}h^2+\cdots+nxh^{n-1}+h^n}h

A power of h in every term of the numerator cancels with h in the denominator:

\displaystyle\lim_{h\to0}\left(nx^{n-1}+\dfrac{n(n-1)}2x^{n-2}h+\cdots+nxh^{n-2}+h^{n-1}\right)

Finally, each term containing h approaches 0 as h\to0, and the derivative is

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4 0
3 years ago
Will give brainliest i rlly need help
Zepler [3.9K]

Answer:

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2 years ago
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olya-2409 [2.1K]

Answer:

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Step-by-step explanation:

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