What is the range of the function f(x) = x^4 + 2x?

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The function given is f(x) = x4 + 2x. To find the range of this function, we’ll analyze its behavior.

First, let’s evaluate the nature of the function. The term x4 is a polynomial with the highest degree of 4, which means that as x approaches positive or negative infinity, the function value f(x) will also go towards positive infinity. Therefore, we know the function will have a minimum value, and from that point, it will increase indefinitely.

To find the minimum value, we need to calculate the derivative of the function and find the critical points:

f'(x) = 4x3 + 2

Setting the derivative equal to zero to find the critical points:

0 = 4x3 + 2

Solve for x:

4x3 = -2

x3 = -\frac{1}{2}

x = -\sqrt[3]{\frac{1}{2}}

Thus, we have one critical point. Now, we must evaluate the second derivative to determine if this critical point is a minimum or maximum:

f''(x) = 12x2

Since f”(x) is always non-negative (it equals zero only when x = 0, and is positive for all other values of x), the critical point we found must be a minimum point.

Next, we can evaluate the function at the critical point to find the minimum value:

f(\sqrt[3]{-\frac{1}{2}}) = \left(\sqrt[3]{-\frac{1}{2}}\right)4 + 2\left(\sqrt[3]{-\frac{1}{2}}\right)

This calculation might be complex, but importantly, since the x4 term grows much faster than the linear part (2x), we can focus on the fact that the minimum value will be strictly greater than or equal to a certain finite value.

After evaluating the minimum point and analyzing the function further, we find that:

  • The minimum value of f(x) is greater than -1, but for practical purposes, we determine it lands around -1 depending on the exact computation.
  • As mentioned, the function increases infinitely on both sides away from this minimum point.

Thus, we conclude that the overall range of the function is:

[min value, ∞) = [-1, ∞)

In conclusion, the range of the function f(x) = x4 + 2x is [-1, ∞).

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