The is the statement of the Mean Value Theorem from your $te\times tb00k$ $Tneoren$ $m4.5$ $Ne8n$ Value Theorem Let f be continuous over $leclose$ interval $\left(a.b\right)aod$ $i$ $eo$ $xd$ $csnlc$ $\left(ab\right)$ $mmd$ exists at least one point cE $\left(a.b\right)$ $si1dhtba$ $f^{'}\left(c\right)=\dfrac {f\left(b\right)-f\left(a\right)} {b-a}$ What is the geometric interpretation of the conclusion of the theorem? O The tangent line to the graph of \(f\left(x\right)\) at $l\left(c1\right)$ is parallel to the secant line connecting \(\left(a,f\left(a\right)\right)\) and \(\left(b,f\left(b\right)\right)\). O The tangent line to the graph of $\left(f\left(lef\left(x\left(right\right)$ at \(c\) is the secant line connecting \(\left(a,f\left(a\right)\right)\) $ana$ \(\left(b,f\left(b\right)\right)\). O The tangent line to the graph $of$ $\left(f\left(leF\left(x\left(right\right)\left(\right)at1\left(c1\right)is$ perpendicular to the secant line connecting \(\left(a,f\left(a\right)\right)\) and A(\left(b,f\left(b\right)\right)\). O The tangent line to the graph of $\left(fleH\left(x\right)nght\right)\right)\right)$ $at$ $\left(c\right)\right)ishoizonta|$ $Tnere$ is more than one tangent line to the graph $0no$ $\left(flcf\left(x\right)night\right)\left($ $at1\left(c1\right)$

$\left(a\right)$ Give $a$ geometrical interpretation of the Rolle's Theorem. $\left(b\right)$ Use the Rolle's Theorem and the fact that $\dfrac {d} {dx}\left(xln\left(2-x\right)\right)=ln\left(2-x\right)$ $-$ $\dfrac {x} {2-x}$ to show that the equation $x=\left(2-x\right)ln\left(2-x\right)$ has at least one solution in the interval $\left(0,1\right)$ $\left(2+3=5$ marks) $1$

Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. $e^{x}=4-3x,$ $\left(0,1\right)$ The equation $e^{x}=4-3x$ is equivalent to the equation $f\left(x\right)=e^{x}-4+3x=0$ $f\left(x\right)$ is continuous on the interval $10,1\right)$ $f\left(0\right)=|$ and $\pi 1\right)=$ Since $Se|eC1-$ $<0<$ < $-Se|ect-$ there is a number $c$ in $\left(0,1\right)$ $5uCH$ that $f\left(c\right)=0$ $by$ the Intermediate Value Theorem. Thus, there is a root of the equation $e^{x}=4-3x,$ $n$ the interval $\dfrac {\bar{1ec1-v} } {\left(0,1\right)}$