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Matematika - LS 24/25

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Spočtěte limitu

\displaystyle\lim_{x\to -3}\frac{x^2+12x+27}{x^2+9x+18}\displaystyle\lim_{x\to -3}\frac{x^2+12x+27}{x^2+9x+18}
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Určete, ve kterém bodě má funkce y=-4x^3+6x^2+72x+5y=-4x^3+6x^2+72x+5 lokální maximum.
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Určete maximální intervaly monotonie funkce

f:y=e^x(-3x+9)f:y=e^x(-3x+9).
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Určete maximální intervaly konvexity a konkávnosti funkce

f:y=-x^3+9x^2-15x+4f:y=-x^3+9x^2-15x+4.
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Spočtěte absolutní minimum (nejmenší funkční hodnotu) funkce f(x)=2x^3+8x^2+6f(x)=2x^3+8x^2+6 pro x\in\langle -8,8\ranglex\in\langle -8,8\rangle.

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Nalezněte asymptoty grafu y= \frac{x^2+11x+28}{9x-7}y= \frac{x^2+11x+28}{9x-7}.
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Spočtěte limitu

\displaystyle\lim_{x\to -5}\frac{x^2+10x+25}{x^2+14x+45}\displaystyle\lim_{x\to -5}\frac{x^2+10x+25}{x^2+14x+45}
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Spočtěte limitu \displaystyle\lim_{x\to\infty} \frac{8x^2+3x+16}{5x^2+8x+15}\displaystyle\lim_{x\to\infty} \frac{8x^2+3x+16}{5x^2+8x+15}

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Vypočtěte směrnici tečny ke grafu funkce f(x)=\sqrt { x^2-3x+5}f(x)=\sqrt { x^2-3x+5} v bodě T=[-6,?].T=[-6,?]. Zaokrouhlete na dvě desetinná místa.

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Jaká musí být hodnota čísla q, aby přímka 28x-4y+q=0 byla tečnou grafu y=4x²-x+7?
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