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# Differentiability

Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.

# Characteristics of graphs of f and f'

Given $$f'(x) = x^2 + 9x + k,$$ if $$f(x)$$ is an increasing function for all values of $$x,$$ what is the range of $$k?$$

Suppose $$f(x)$$ is a function such that $$y=f'(x)$$ is given by the curve in the above diagram. If $$y= f'(x)$$ is tangent to the $$x$$-axis at $$x=a=7,$$ which of the following statements is NOT correct about $$f(x)?$$

Suppose $$f(x)$$ is a differentiable function such that the graph of $$y=f'(x)$$ is as shown in the above diagram. If $a = 2, b = 3, c = 5, d= 7,$ and $$f'(e) = f'(d)$$ for all $$e \geq d,$$ which of following statements is correct?

Suppose $$f(x)$$ is a function such that the graph of $$y=f'(x)$$ is a line as shown in the above diagram. If $$a=19,$$ which of following statements is correct?

Suppose $$f(x)$$ is a function such that $$f'(x) = K(x-a)(x-b) ,$$ where $$K$$ is a positive constant. If $$a=5$$ and $$b=10,$$ which of following statements is NOT correct?

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