Calculus › Extrema and curve sketching

Bounds from derivatives

Solve the first problem 9 problems

1. Most economical car speed ●○○○○ 10 ptsA car's fuel use is given by $f(v)=\frac{3}{80}v^{2}-6v+245$, where $v$ is the travelling speed. Find the speed $v$ that minimises the fuel use.
2. Minimum of f(m) + f'(n) on an interval ●●○○○ 20 ptsThe function $f(x) = -x^3 + ax^2 - 4$ has a local extremum at $x = 2$. For $m, n \in [-1,1]$, find the minimum value of $f(m) + f'(n)$, where $f'$ is the derivative of $f$.
3. Smallest value from derivative bounds ●●○○○ 20 ptsGiven that $f(1) = 3$ and $2 \leq f'(x) \leq 3$ for all $x$, what is the smallest possible value of $f(4)$?
4. Smoothness of f from an integral ●●○○○ 20 ptsLet $f$ be a continuous function and let $g(x) = \int_0^x f(y)(y - x)\,dy$. Suppose $g$ is three times continuously differentiable. What is the greatest integer $n$ such that $f$ must be $n$ times continuously differentiable?
5. Smallest bound on derivative at zero ●●●○○ 50 ptsFind the smallest real number $A$ such that for every quadratic polynomial $f(x)$ satisfying $|f(x)| \leqslant 1$ for all $0 \leqslant x \leqslant 1$, the inequality $f'(0) \leqslant A$ holds.
6. Minimum roots of g times g' ●●●○○ 50 ptsLet $f$ be a function defined for all $x \in \mathbb{R}$ that is discontinuous at exactly $3$ values of $x$. Let $g$ be a differentiable function such that $f(x)g(x)$ is continuous for all $x \in \mathbb{R}$. Find the minimum number of distinct real roots of the equation $g(x)g'(x)=0$.
7. Smallest bound on function differences ●●●○○ 50 ptsThe function $f(x)$ is defined on $[0,1]$ and satisfies $f(0)=f(1)$, and for any $x, y \in [0,1]$ with $x \neq y$, we have $|f(x)-f(y)|<|x-y|$. Find the smallest real number $m$ such that for every such function $f$ and any $x, y \in [0,1]$, we have $|f(x)-f(y)|<m$. Give your answer as a decimal.
8. Maximum integer k in a derivative inequality ●●●●○ 100 ptsLet $f(x)=e^{x}-x-2$. For $x>0$, the inequality $(x-k)f'(x)+x+1>0$ always holds, where $k$ is an integer. Find the maximum value of $k$.
9. Minimum ratio of derivative energy to squared mean ●●●●○ 100 ptsLet $f$ be a continuously differentiable real-valued function on $[0,1]$ with $\int_{1/3}^{2/3} f(x)\,dx = 0$. Find the minimum value of $\dfrac{\int_0^1 (f'(x))^2\,dx}{\left(\int_0^1 f(x)\,dx\right)^2}$ over all such $f$ for which the denominator is nonzero.

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