The interval in which the function $ f(x)=\sin(e^x)+\cos(e^x)$ is increasing is/are?

Question

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The interval in which the function $f(x)=\sin(e^x)+\cos(e^x)$ is increasing is/are?

I don't understand how to approach such problems. it would be helpful if you could kindly guide me through the process. i have also shared the options image and the correct answers have a green tick.

Answer 1

Hint:

$$\sin y+\cos y=\sqrt2\sin\left(y+\frac\pi4\right)$$ is growing in $$\left[-\frac{3\pi}4,\frac{\pi}4\right]+2k\pi$$

and the transformation

$$y=e^x$$ is invertible.

Answer 2

$$f^{'}(x)=e^x(\sin e^x - \cos e^x)>0$$ gives (first letter in alphabet)