Determine os intervalos de crescimento e decrescimento da função:
F: R --> R
x |--> f(x) = x² - x - 2
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Olá!
Basta analisar a primeira derivada. Ela dará os intervalos de crescimento e decrescimento.
![f(x)=x^2-x-2\Rightarrow f'(x)= 2x-1\\ \\ \text{Agora, }\\ \\ f'(x)=0\Leftrightarrow 2x-1=0\Leftrightarrow x=\dfrac{1}{2}.\\ \\ \\ \text{Logo,}\\
\\ f'(x)\ \textgreater \ 0\Leftrightarrow x\in\left]\dfrac{1}{2},+\infty\right[\\ \\ \text{e}\\ \\
f'(x)\ \textless \ 0\Leftrightarrow x\in\left]-\infty,\dfrac{1}{2}\right[\\ \\ \text{Portanto, $f$ \'e estritamente crescente em }\\ \\ \left]\dfrac{1}{2},+\infty\right[\\ \\ \text{e $f$ \'e estritamente decrescente em}\\ \\ \left]-\infty,\dfrac{1}{2}\right[](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E2-x-2%5CRightarrow+f%27%28x%29%3D+2x-1%5C%5C+%5C%5C+%5Ctext%7BAgora%2C+%7D%5C%5C+%5C%5C+f%27%28x%29%3D0%5CLeftrightarrow+2x-1%3D0%5CLeftrightarrow+x%3D%5Cdfrac%7B1%7D%7B2%7D.%5C%5C+%5C%5C+%5C%5C+%5Ctext%7BLogo%2C%7D%5C%5C%0A%5C%5C+f%27%28x%29%5C+%5Ctextgreater+%5C+0%5CLeftrightarrow+x%5Cin%5Cleft%5D%5Cdfrac%7B1%7D%7B2%7D%2C%2B%5Cinfty%5Cright%5B%5C%5C+%5C%5C+%5Ctext%7Be%7D%5C%5C+%5C%5C+%0Af%27%28x%29%5C+%5Ctextless+%5C+0%5CLeftrightarrow+x%5Cin%5Cleft%5D-%5Cinfty%2C%5Cdfrac%7B1%7D%7B2%7D%5Cright%5B%5C%5C+%5C%5C+%5Ctext%7BPortanto%2C+%24f%24+%5C%27e+estritamente+crescente+em+%7D%5C%5C+%5C%5C+%5Cleft%5D%5Cdfrac%7B1%7D%7B2%7D%2C%2B%5Cinfty%5Cright%5B%5C%5C+%5C%5C+%5Ctext%7Be+%24f%24+%5C%27e+estritamente+decrescente+em%7D%5C%5C+%5C%5C++%5Cleft%5D-%5Cinfty%2C%5Cdfrac%7B1%7D%7B2%7D%5Cright%5B "f(x)=x^2-x-2\Rightarrow f'(x)= 2x-1\\ \\ \text{Agora, }\\ \\ f'(x)=0\Leftrightarrow 2x-1=0\Leftrightarrow x=\dfrac{1}{2}.\\ \\ \\ \text{Logo,}\\
\\ f'(x)\ \textgreater \ 0\Leftrightarrow x\in\left]\dfrac{1}{2},+\infty\right[\\ \\ \text{e}\\ \\
f'(x)\ \textless \ 0\Leftrightarrow x\in\left]-\infty,\dfrac{1}{2}\right[\\ \\ \text{Portanto, $f$ \'e estritamente crescente em }\\ \\ \left]\dfrac{1}{2},+\infty\right[\\ \\ \text{e $f$ \'e estritamente decrescente em}\\ \\ \left]-\infty,\dfrac{1}{2}\right[")
Bons estudos!
Basta analisar a primeira derivada. Ela dará os intervalos de crescimento e decrescimento.
Bons estudos!
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