Calculus

Differentiability

Rolle's Theorem

         

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

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

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What is the value of lim⁡x→0+e7sin⁡x−e7xsin⁡x−x?\displaystyle{\lim_{x \rightarrow 0^{+} }\frac{e^{7\sin x}-e^{7x}}{\sin x - x} }?

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How many real solutions are there to the following equation: 43x3+7x+2=0? \frac{4}{3}x^3 + 7x + 2=0?

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Suppose f(x)f(x) is a differentiable function on [6,13]\left [ 6, 13 \right ] such that (2x−19)(f(x)−6)≠f′(x)(x2−19x+78) (2x-19)(f(x)-6) \neq f'(x)(x^2 - 19x + 78) for any x∈[6,13].x \in \left [ 6, 13 \right ]. Which of the following values must be contained in f([6,13])?f(\left [ 6, 13 \right ])?

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