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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