Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative$$- \frac{2 \left(2 x + \left(1 - \frac{\left(2 x + 7\right)^{2}}{x \left(x + 7\right)}\right) \left(x - 3\right) + 7\right) \log{\left(5 \right)}}{x^{2} \left(x + 7\right)^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 3 + \sqrt[3]{90} + \sqrt[3]{300}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -7$$
$$x_{2} = 0$$
$$\lim_{x \to -7^-}\left(- \frac{2 \left(2 x + \left(1 - \frac{\left(2 x + 7\right)^{2}}{x \left(x + 7\right)}\right) \left(x - 3\right) + 7\right) \log{\left(5 \right)}}{x^{2} \left(x + 7\right)^{2}}\right) = -\infty$$
$$\lim_{x \to -7^+}\left(- \frac{2 \left(2 x + \left(1 - \frac{\left(2 x + 7\right)^{2}}{x \left(x + 7\right)}\right) \left(x - 3\right) + 7\right) \log{\left(5 \right)}}{x^{2} \left(x + 7\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -7$$
- is an inflection point
$$\lim_{x \to 0^-}\left(- \frac{2 \left(2 x + \left(1 - \frac{\left(2 x + 7\right)^{2}}{x \left(x + 7\right)}\right) \left(x - 3\right) + 7\right) \log{\left(5 \right)}}{x^{2} \left(x + 7\right)^{2}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(- \frac{2 \left(2 x + \left(1 - \frac{\left(2 x + 7\right)^{2}}{x \left(x + 7\right)}\right) \left(x - 3\right) + 7\right) \log{\left(5 \right)}}{x^{2} \left(x + 7\right)^{2}}\right) = -\infty$$
- the limits are not equal, so
$$x_{2} = 0$$
- is an inflection point
Сonvexity and concavity intervals:Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[3 + \sqrt[3]{90} + \sqrt[3]{300}, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 3 + \sqrt[3]{90} + \sqrt[3]{300}\right]$$