Let $f : [a, b] \to R$ be continuous and differentiable on $(a, b)$ . Darboux's theorem on derivatives states that:

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If $x^{2} + y^{2} = 25$ , what is the second derivative $\frac{d ^{2} y}{d x ^{2}}$ at the point $(3, 4)$ ?
Let $f$ be twice differentiable on $R$ with $f^{''} (x) > 0$ everywhere. Which statement about the tangent lines is true?
What is the derivative of $f (x) = ln (sec x + tan x)$ for $x \in (- π /2, π /2)$ ?
On which interval is the function $f (x) = x - ln x$ strictly increasing?
If $f$ is differentiable on $[0, 2]$ with $f (0) = 1$ and $f^{'} (x) \leq 3$ for all $x \in (0, 2)$ , what is the maximum possible value of $f (2)$ ?
How many inflection points does the polynomial $f (x) = x^{4} - 6 x^{2} + 5$ possess?

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