Answer:
Yes
Step-by-step explanation:
Answer:
c
Step-by-step explanation:
![i^5+i^{-25}+i^{45}](https://tex.z-dn.net/?f=i%5E5%2Bi%5E%7B-25%7D%2Bi%5E%7B45%7D)
Rewrite the term ![i^{-25}](https://tex.z-dn.net/?f=i%5E%7B-25%7D)
![=i^5+\dfrac{1}{i^{25}}+i^{45}](https://tex.z-dn.net/?f=%3Di%5E5%2B%5Cdfrac%7B1%7D%7Bi%5E%7B25%7D%7D%2Bi%5E%7B45%7D)
Expand each term so we have
![=i(i^2)^2+\dfrac{1}{i(i^2)^{12}}+i(i^2)^{22}](https://tex.z-dn.net/?f=%3Di%28i%5E2%29%5E2%2B%5Cdfrac%7B1%7D%7Bi%28i%5E2%29%5E%7B12%7D%7D%2Bi%28i%5E2%29%5E%7B22%7D)
Use the fact that ![i^2=-1](https://tex.z-dn.net/?f=i%5E2%3D-1)
![=i(-1)^2+\dfrac{1}{i(-1)^{12}}+i(-1)^{22}](https://tex.z-dn.net/?f=%3Di%28-1%29%5E2%2B%5Cdfrac%7B1%7D%7Bi%28-1%29%5E%7B12%7D%7D%2Bi%28-1%29%5E%7B22%7D)
Use the fact that
when a is an even number
![=i+\dfrac{1}{i}+i](https://tex.z-dn.net/?f=%3Di%2B%5Cdfrac%7B1%7D%7Bi%7D%2Bi)
Simplify
![=i-i+i](https://tex.z-dn.net/?f=%3Di-i%2Bi)
![=i](https://tex.z-dn.net/?f=%3Di)
Let me know if you need any clarifications, thanks!
Answer: The second derivative of the function is ![f''(x)=\dfrac{1}{2x}](https://tex.z-dn.net/?f=f%27%27%28x%29%3D%5Cdfrac%7B1%7D%7B2x%7D)
Step-by-step explanation:
Since we have given that
![f(x)=x\ln \sqrt{x}+2x](https://tex.z-dn.net/?f=f%28x%29%3Dx%5Cln%20%5Csqrt%7Bx%7D%2B2x)
We need to find the second derivative of the function.
So, the first derivative would be
![f'(x)=1\times \ln\sqrt{x}+x\dfrac{1}{\sqrt{x}}\times \dfrac{1}{2\sqrt{x}}+2\\\\f'(x)=\ln \sqrt{x}+\dfrac{1}{2}+2\\\\f'(x)=\ln\sqrt{x}+\dfrac{3}{2}](https://tex.z-dn.net/?f=f%27%28x%29%3D1%5Ctimes%20%5Cln%5Csqrt%7Bx%7D%2Bx%5Cdfrac%7B1%7D%7B%5Csqrt%7Bx%7D%7D%5Ctimes%20%5Cdfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%2B2%5C%5C%5C%5Cf%27%28x%29%3D%5Cln%20%5Csqrt%7Bx%7D%2B%5Cdfrac%7B1%7D%7B2%7D%2B2%5C%5C%5C%5Cf%27%28x%29%3D%5Cln%5Csqrt%7Bx%7D%2B%5Cdfrac%7B3%7D%7B2%7D)
Now, second derivative would be
![f''(x)=\dfrac{1}{\sqrt{x}}\times \dfrac{1}{2\sqrt{x}}\\\\f''(x)=\dfrac{1}{2x}](https://tex.z-dn.net/?f=f%27%27%28x%29%3D%5Cdfrac%7B1%7D%7B%5Csqrt%7Bx%7D%7D%5Ctimes%20%5Cdfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%5C%5C%5C%5Cf%27%27%28x%29%3D%5Cdfrac%7B1%7D%7B2x%7D)
Hence, the second derivative of the function is ![f''(x)=\dfrac{1}{2x}](https://tex.z-dn.net/?f=f%27%27%28x%29%3D%5Cdfrac%7B1%7D%7B2x%7D)
The integer is -56. The word “lost” represents negative.