Let q be a function that is continuous on [0,k] and differentiable on (0,k) for every k>0. Suppose the derivative of q is always bounded between −2 and 6 on that interval:
−2≤q′(x)≤6for 0≤x≤k.
Assume that q(0)=5.
a. Express upper and lower bounds for q(k) in terms of k. Provide complete justification for your bounds.
b. Using the bounds from part (a), determine the greatest integer value of k (if any) for which q(k)≤25 is guaranteed. Explain your reasoning clearly.
c. Give an explicit numerical example of a value of k and a differentiable function q (defined on [0,k] and satisfying the derivative bounds) that attains your upper bound for q(k) from part (a). Justify that your example meets all required conditions.