本题目来源于试卷: Differential Calculus，类别为 IB数学
[问答题]
Let $f(x)=\frac{1}{3} x^{3}+2 x^{2}-5 x+10$ .
1. Find $f^{\prime}(x)$ .
The graph of f has horizontal tangents at the points where x=a and x=b, a$\lt$b .
2. Find the value of a and the value of b .
3. a. Sketch the graph of $y=f^{\prime}(x)$ .
b. Hence explain why the graph of f has a local maximum point at x=a .
4. a. Find $f^{\prime \prime}(b)$ .
b. Hence, use your answer to part (d) (i) to show that the graph of f has a local minimum point at x=b .
The tangent to the graph of f at x=a and the normal to the graph of f at x=b intersect at the point (p, q) .
5. Find the value of p and the value of q .
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