Calculus

# Rolle's Theorem

Suppose $f$ and $g$ are differentiable functions on the interval $[ 8 , 42 ]$ such that $f \left( 8\right) = g \left( 8\right)$ and $f' \left( x \right) < g' \left( x \right)$ for all $x$ on $( 8 , 42 ) .$ If $f \left( 42\right) = 48$ , which of the following values NOT possible for $g \left( 42\right)?$

Suppose $f(x)$ is a differentiable function on the interval $\left [6, 9 \right ]$ such that $f(6) = 18$ and $f(9) = 27 .$ Which of the following values must be contained in $f'(\left [6, 9 \right ])?$

What is the value of $\displaystyle{\lim_{x \rightarrow 0^{+} }\frac{e^{7\sin x}-e^{7x}}{\sin x - x} }?$

How many real solutions are there to the following equation: $\frac{4}{3}x^3 + 7x + 2=0?$

Suppose $f(x)$ is a differentiable function on $\left [ 6, 13 \right ]$ such that $(2x-19)(f(x)-6) \neq f'(x)(x^2 - 19x + 78)$ for any $x \in \left [ 6, 13 \right ].$ Which of the following values must be contained in $f(\left [ 6, 13 \right ])?$

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