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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.

Graphs of Derivatives

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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