For x E [-13, 15] the function f is defined by On which two intervals is the fun

Question

For x E [-13, 15] the function f is defined by

On which two intervals is the function increasing?
..... to .....
and
.....to 15
Find the region in which the function is positive: 4 to 15
Where does the function achieve its minimum?

I got 16/7 but have inserted in different parts of the question but I got it wrong everytime. For the region I got 4-15 and for the increasing intervals..the last interval I got ..to 15. I got -1 for the minimum but it was wrong.

Help with detailed explanation please!

Explanation / Answer

(f(x) = x^4(x-4)^5) [-13, 15] First find the derivative via the product rule.

(f'(x) = 5x^4(x - 4)^4 + 4x^3(x - 4)^5)

(f'(x) = x^3(x - 4)^4[5x + 4(x - 4)] = x^3(x-4)^4[9x - 16])

Setting this equal to 0, you get: x = 0, x = 4 and x = 16/9

We need to consider the following intervals:

[-13, 0), (0, 16/9) , (16/9, 4), (4, 15]

f(x) is increasing where f'(x) > 0. This happen on:

[-13, 0) , (16/9, 4) and (4, 15]

So that's actually 3 intervals on which f(x) is increasing.

f(x) will be positive when (x - 4) > 0, x > 4, so it's : (4, 15]

For the minimum and maximum, test all of the points.

The minimum occurs at x = -13. All you needed to do is evaluate

f(x) at the critical points and the endpoints to get that.

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