16 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 ] .

E377

[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’]

E377

[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’]

E377

[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’]

E377

[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’]

E377

[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’]

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

E377

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

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[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’]

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[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’]

E377

[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’]