4.3 Consequences of differentiability, the mean value theorem  (Page 3/3)

( ${cos}^2+{sin}^2=1$ and $exp(i\pi =-1.$ )

1. Prove that ${cos}^2\left(z\right)+{sin}^2\left(z\right)=1$ for all complex numbers $z.$
2. Prove that $cos\left(\pi \right)=-1.$ HINT: We know from part (a) that $cos\left(\pi \right)=\pm 1.$ Using the Mean Value Theorem for the cosine function on the interval $[0,\pi ],$ derive a contradiction from the assumption that $cos\left(\pi \right)=1.$
3. Prove that $exp\left(i\pi \right)=-1.$ HINT: Recall that $exp\left(iz\right)=cos\left(z\right)+isin\left(z\right)$ for all complex $z.$ (Note that this does not yet tell us that $e^{i\pi }=-1.$ We do not yet know that $exp\left(z\right)=e^z.$ )
4. Prove that ${cosh}^{2}z-{sinh}^{2}z=1$ for all complex numbers $z.$
5. Compute the derivatives of the tangent and hyperbolic tangent functions $tan=sin/cos$ and $tanh=sinh/cosh.$ Show in fact that
   $${tan}^{\text{'}}=\frac{1}{{cos}^2}\text{and}{tanh}^{\text{'}}=\frac{1}{{cosh}^2}.$$

1. Suppose $f$ and $g$ are two complex-valued functions of a real (or complex) variable,and suppose that $f^{\text{'}}\left(x\right)=g^{\text{'}}\left(x\right)$ for all $x\in (a,b)$ (or $x\in B_r\left(c\right).$ ) Prove that there exists a constant $k$ such that $f\left(x\right)=g\left(x\right)+k$ for all $x\in (a,b)$ (or $x\in B_r\left(c\right).$ )
2. Suppose $f^{\text{'}}\left(z\right)=cexp\left(az\right)$ for all $z,$ where $c$ and $a$ are complex constants with $a\ne 0.$ Prove that there exists a constant $c^{\text{'}}$ such that $f\left(z\right)=\frac{c}{a}exp\left(az\right)+c^{\text{'}}.$ What if $a=0?$
3. (A generalization of part (a)) Suppose $f$ and $g$ are continuous real-valued functions on the closed interval $[a,b],$ and suppose there exists a partition $\{x_0<x_1<...<x_n\}$ of $[a,b]$ such that both $f$ and $g$ are differentiable on each subinterval $(x_{i-1},x_i).$ (That is, we do not assume that $f$ and $g$ are differentiable at the endpoints.) Suppose that $f^{\text{'}}\left(x\right)=g^{\text{'}}\left(x\right)$ for every $x$ in each open subinterval $(x_{i-1},x_i).$ Prove that there exists a constant $k$ such that $f\left(x\right)=g\left(x\right)+k$ for all $x\in [a,b].$ HINT: Use part (a) to conclude that $f=g+h$ where $h$ is a step function, and then observe that $h$ must be continuous and hence a constant.
4. Suppose $f$ is a differentiable real-valued function on $(a,b)$ and assume that $f^{\text{'}}\left(x\right)\ne 0$ for all $x\in (a,b).$ Prove that $f$ is 1-1 on $(a,b).$

Let $f:[a,b]\to R$ be a function that is continuous on its domain $[a,b]$ and differentiable on $(a,b).$ (We do not suppose that $f^{\text{'}}$ is continuous on $(a,b).$ )

1. Prove that $f$ is nondecreasing on $[a,b]$ if and only if $f^{\text{'}}\left(x\right)\ge 0$ for all $x\in (a,b).$ Show also that $f$ is nonincreasing on $[a,b]$ if and only if $f^{\text{'}}\left(x\right)\le 0$ for all $x\in (a,b).$
2. Conclude that, if $f^{\text{'}}$ takes on both positive and negative values on $(a,b),$ then $f$ is not 1-1. (See the proof of [link] .)
3. Show that, if $f^{\text{'}}$ takes on both positive and negative values on $(a,b),$ then there must exist a point $c\in (a,b)$ for which $f^{\text{'}}\left(c\right)=0.$ (If $f^{\text{'}}$ were continuous, this would follow from the Intermediate Value Theorem. But, we are not assuming here that $f^{\text{'}}$ is continuous.)
4. Prove the Intermediate Value Theorem for Derivatives: Suppose $f$ is continuous on the closed bounded interval $[a,b]$ and differentiable on the open interval $(a,b).$ If $f^{\text{'}}$ attains two distinct values $v_1=f^{\text{'}}\left(x_1\right)<v_2=f^{\text{'}}\left(x_2\right),$ then $f^{\text{'}}$ attains every value $v$ between $v_1$ and $v_2.$ HINT: Suppose $v$ is a value between $v_1$ and $v_2.$ Define a function $g$ on $[a,b]$ by $g\left(x\right)=f\left(x\right)-vx.$ Now apply part (c) to $g.$