17 Differentiable functions[1C5]

To address the following exercises, it may be necessary to know some fundamental results in Analysis and Differential Calculus that may be found e.g. in [ 22 , 4 ] ; specifically:

• Lagrange’s Theorem  ¹  : Theorem 5.10 in in [ 22 ] , or [ 61 ] .

• De l’Hôpital’ rule, and corollaries: : Theorem 5.13 in in [ 22 ] , Sec. 7.12 in [ 4 ] or [ 24 , 59 ] ;

• Taylor’s Theorem, and the possible remainders: Theorem 5.15 in in [ 22 ] , Chap. 7 in [ 4 ] , or [ 66 ] .

E378

[1C6] Let \(I⊆ ℝ\) be an open interval. Let \(f:I→ℝ\) be differentiable, and \(x,y∈ I\) with \(x{\lt}y\). Show that if \(f'(x)⋅ f'(y){\lt}0\) then \(𝜉∈ I\) exists with \(x{\lt}𝜉{\lt}y\) such that \(f'(𝜉)=0\). Hidden solution: [UNACCESSIBLE UUID ’1C7’]

E378

[1C8] Prerequisites:1.Note:Darboux properties.

Let \(A⊆ ℝ\) be an open set, and suppose that \(f:A→ℝ\) is differentiable. We want to show that, for each interval \(I⊂ A\), the image \(f'(I)\) is an interval.

So prove this result. For \(x,y∈ I\) with \(x{\lt}y\), let’s define \(a=f'(x), b= f'(y)\). Let’s assume for simplicity that \(a{\lt}b\). For any \(c\) with \(a{\lt} c {\lt} b\), there exists \(𝜉∈ I\) with \(x{\lt}𝜉{\lt}y\) such that \(f'(𝜉)=c\).

(Finally, show that this property actually implies that the image \(f'(I)\) of an interval \(I\) is an interval.)

Hidden solution: [UNACCESSIBLE UUID ’1C9’]

E378

[1CB] Prerequisites:2.

Let \(I⊆ ℝ\) be an open interval. Let \(f:I→ℝ\) be a differentiable function such that \(f'(t)≠ 0\) for every \(t∈ I\): show then that \(f'(t)\) has always the same sign.

Hidden solution: [UNACCESSIBLE UUID ’1CC’]

E378

[1CD] Prerequisites:2.Difficulty:*.

Find a bounded function \(f:ℝ→ℝ\) that maps intervals into intervals, but such that there does not exist \(g:ℝ→ℝ\) differentiable at every point and with \(f=g'\).

(Note that \(f\) cannot be continuous, due to the Fundamental Theorem of Calculus.)

Hidden solution: [UNACCESSIBLE UUID ’1CF’]

E378

[1CG]Suppose that \(f:ℝ→ℝ\) be differentiable, with \(f'=f\): prove, in an elementary way, that that there exists \(𝜆∈ℝ\) s.t. \(f(x) = 𝜆 e^ x\). Hidden solution: [UNACCESSIBLE UUID ’1CH’]

E378

[1CJ] Find a differentiable function \(f:ℝ→ℝ\) whose derivative is bounded but not continuous. Hidden solution: [UNACCESSIBLE UUID ’1CK’]

E378

[1CM]Find a continuous and differentiable function \(f:[-1,1]→ℝ\) ² whose derivative is unbounded. Hidden solution: [UNACCESSIBLE UUID ’1CN’]

E378

[1CP] Difficulty:*. Describe a function \(f:ℝ→ℝ\) that is differentiable and such that the image of \([0,1]\) using \(f'\) is \(f'([0,1])=(-1,1)\).

Before looking for the example, ponder on this notions. We remember the Darboux property 2: the image \(f'(I)\) of an interval \(I\) is an interval; but this does not say that the image of \(f'([0,1])\) should be a closed and bounded interval. If, however, we also knew that \(f'\) is continuous, what could we say of \(f'([0,1])\)? So what do you deduce a priori about the sought example?

Hidden solution: [UNACCESSIBLE UUID ’1CQ’]

E378

[1CV]Let \(I=(a,b)⊂ℝ\) be an open interval. Let \(f:I→ℝ\) be differentiable: show that \(f'\) is continuous, if and only if for every \(x\)

\[ f'(x) = \lim _{(s,t)→ (x,x), s≠ t} \frac{f(t)-f(s)}{t-s} ~ . \]

Hidden solution: [UNACCESSIBLE UUID ’1CW’]

E378

[1CX]Let f be differentiable in the interval \((a, b)\), let \(x_ 0 ∈ (a, b)\) and \(x_ 0 {\lt} 𝛼_ n {\lt} 𝛽_ n , 𝛽_ n → x_ 0\) for \(n →∞\). Show that if the sequence \(\frac{𝛽_ n - x_ 0}{𝛽_ n - 𝛼_ n}\) is bounded then

\[ \frac{f (𝛽_ n ) - f (𝛼_ n )}{𝛽_ n - 𝛼_ n}→_ n f' (x_ 0 ) \]

Show by example that this conclusion is false if the given condition is not verified. [UNACCESSIBLE UUID ’1CY’]

[1CZ]Suppose that a given function \(f : (a, b) → R\) is differentiable at every point of \((a, b)\) except \(x_ 0\), and that the limit \(\lim _{t→ x_ 0} f (t)\) exists and is finite. Show that f is also differentiable in \(x_ 0\) and that \(f (x_ 0 ) = \lim _{t→ x_ 0} f (t)\).

[UNACCESSIBLE UUID ’1D0’] [1D1]Prerequisites:2.Let \(f, g : ℝ → ℝ\) be two functions that can be differentiated at every point. Show that \(\max \{ f, g\} \) is differentiable, except on a set that is at most countable. Hidden solution: [UNACCESSIBLE UUID ’1D2’][UNACCESSIBLE UUID ’1D3’] [1D4]Let \(f : (a, b) → ℝ\) be differentiable and such that, if \(f (t) = 0\), then \(f' (t) = 0\). Show that the function \(g(t) = |f (t)|\) is differentiable. [UNACCESSIBLE UUID ’1D5’] Hidden solution: [UNACCESSIBLE UUID ’1D6’] [1D7]Let \(p(x) = a_ n x^ n + a_{n-1} x^{n-1} + . . . + a_ 0\) a polynomial with all real roots and coefficients all non-zero. Show that the number of positive roots (counted with multiplicity) is equal to the number of sign changes in the sequence of coefficients of \(p\). [Hint. Use induction on \(n\), using the fact that between two consecutive roots of \(p\) there exists a root of \(p'\) .] This result is known as Descartes’ rule of signs. [UNACCESSIBLE UUID ’1D8’] [1D9]Let \(f:ℝ→ℝ\) be continuous and differentiable, and \(a,b∈ℝ\) with \(a{\lt}b\). Show that, if \(f'(a)=f'(b)\), then \(𝜉\) exists with \(a{\lt}𝜉{\lt}b\) such that

[UNACCESSIBLE UUID ’1DB’] [UNACCESSIBLE UUID ’1DC’]