Question

2^x+5^2x-sinx=e^x x=?

Answer 1

$$2^{x} + 5^{2x} - \sin x = e^{x}$$

It can't be solved by standard methods.

Let $f(x) = 2^x + 5^{2x} - \sin x - e^x$, $f'(x) = 2^x \ln 2 + 2 \cdot 5^{2x} \ln 5 - \cos x - e^x$

$$f(-3) = 0.224$$

$f(-4) = -0.711$ – it shows that our root lays between -3 and -4.

Use a Newton's method

$$a_1 = a - f(a) \frac{b - a}{f(b) - f(a)}$$ $$x_1 = b - \frac{f(b)}{f'(b)}$$

$f''(x) > 0$ when $x \in [-3, -4]$ and $f(-3) > 0$ then $b = -3$, $a = -4$.

$$x_1 = -3 - \frac{f(-3)}{f'(-3)} = -3.216$$ $$a_1 = -4 - f(-4) \frac{-3 + 4}{f(-3) - f(-4)} = -3.24$$ $$x_2 = -3.216 - \frac{f(-3.216)}{f'(-3.216)} = -3.215$$ $$a_1 = -3.24 - f(-3.24) \frac{-3.216 + 3.24}{f(-3.216) - f(-3.24)} = -3.215$$

Consequently $x \approx -3.215$.