# For f''(x) = (x-6)^5 (x-2)^3 (x+5)^7 a) f(x) has an inflection point at x = -5 and has an...

## Question:

For {eq}f''(x) = (x-6)^5 (x-2)^3 (x+5)^7 {/eq}

a) {eq}f(x) {/eq} has an inflection point at {eq}x = -5 {/eq} and has an inflection point at {eq}x = 2 {/eq}. Explain why that statement is true.

b) Also, what is the rule that will determine if {eq}f'(x) {/eq} has a maximum/minimum ?

## Inflection Points and Maximum/Minimum of a Function

Given the second derivative of a function f(x) we comment on its inflection points. Then we state the criteria necessary for the function to have a relative maximum and relative minimum at points(s) in its domain. The concepts are those from Calculus and involve the first and second derivatives as well as zeros of functions.

## Answer and Explanation:

a) Since {eq}f''(-5)=0 {/eq} and {eq}f''(2)=0, {/eq} therefore {eq}x=-5 \; {\rm and} \; x=2 {/eq} are both inflection points for {eq}f(x). {/eq}

b) From Calculus, {eq}f'(x) {/eq} will have a relative maximum at point {eq}x=a {/eq} in its domain if {eq}f''(a)=0 \; {\rm and} \; f'''(a)<0. {/eq}

Similarly, {eq}f'(x) {/eq} will have a relative minimum at point {eq}x=a {/eq} in its domain if {eq}f''(a)=0 \; {\rm and} \; f'''(a)>0. {/eq}

There is another way a function can have a relative maximum or minimum at a point as follows:

{eq}f'(x) {/eq} may have a relative maximum or relative minimum at point {eq}x = a {/eq} in its domain if {eq}f''(a) {/eq} does not exist or is UNDEFINED. In that case one would need to use graphical techniques to determine whether a relative maximum or relative minimum exists at this point. This, since the second derivative test will no longer work owing to higher order derivatives NOT defined either.