Find the largest and smallest value of the function f (x) = x ^ 4 -2x on the segment [0.3].

Given a function:

f (x) = x ^ 4 – 2x;

Let’s find the derivative of this function:

f ‘(x) = 4 * x ^ 3 – 2 * 1 = 4x ^ 3 – 2.

Let’s find the zeros of this derivative on the graphs, equating it to zero:

4x ^ 3 – 2 = 0;

4x ^ 3 = 2;

x ^ 3 = 2/4;

x = ^ 3√2/4.

Now let’s substitute this value into the function:

f (^ 3√ 2/4) = (^ 3√ 2/4) ^ 4 – 2 * ^ 3√ 2/4 = – (3 * ^ 3√ 4) / 4 ~ (approximately equal) ~ -1.2.

Substitute the extreme points of the segment that this function belongs to:

f (0) = 0 ^ 4 – 2 * 0 = 0 – 0 = 0.

f (3) = 3 ^ 4 – 2 * 3 = 81 – 6 = 75.

Thus, the smallest value of this function on the segment [0; 3]: -1.2.

And the largest value of the function on the segment [0; 3] = 75.

Answer: highest value: 75; smallest value: -1.2.