Chapter 4 - Applications of the Derivative - 4.4 The Shape of a Graph - Exercises - Page 194: 10

concave up $x$ > $4$ concave down $0$ < $x$ < $4$ inflection point is at $x$ = $4$

let $f(x)$ = $x^{\frac{7}{2}}-35x^{2}$ $f'(x)$ = $\frac{7}{2}x^{\frac{5}{2}}-70x$ $f''(x)$ = $\frac{35}{4}x^{\frac{3}{2}}-70$ concave up $\frac{35}{4}x^{\frac{3}{2}}-70$ > $0$ $x^{\frac{3}{2}}$ > $8$ $x$ > $4$ concave down $0$ < $x$ < $4$ inflection point is at $x$ = $4$