Question

Estimate the x-coordinates at which the relative maxima and relative minima occur for the function.

Answer 1

ANSWER

Relative minimum occurs at x=0

Relative maximum occurs at x=-2

EXPLANATION

The given function is;

$f(x) = 3 {x}^{3} + 9 {x}^{2} - 1$

We take the first derivative to get:

$f'(x) = 9 {x}^{2} + 18x$

At turning point f'(x)=0

$9 {x}^{2} + 18x = 0$

$9x(x + 2) = 0$

$x = 0 \: or \: x = - 2$

We take the second derivative to get

$f''(x) = 18x + 18$

$f''(0) = 18(0) + 18 = 18 \: > \: 0$

Relative minimum occurs at x=0

$f''( - 2) = 18( - 2)+ 18 = - 18 \: < \: 0$

Relative maximum occurs at x=-2

Answer 2

Answer:

Relative minimum occurs at x=0

Relative maximum occurs at x=-2

Step-by-step explanation: