Previous Problem Problem List Next Problem Suppose that \[ f(x)=2 x^{3}+5 x \] (A) Find all critical values of $f$. If there are no critical values, enter -1000 . If there are more than one, enter them separated by commas. Critical value(s) $=$ (B) Use interval notation to indicate where $f(x)$ is increasing. Note: When using interval notation in WeBWork, you use I for $\infty,-I$ for $-\infty$, and $U$ for the union symbol, If there are no values that satisfy the required condition, then enter " 0 " without the quotation marks. Increasing: (C) Find the z-coordinates of all local maxima of $f$. If there are no local maxima, enter-1000. If there are more than one, enter thern separated by commas. Local maxima at $\mathbf{z}=$ (D) Find the $z$-coordinates of all local minima of $f$. If there are no local minima, enter - 1000 . If there are more than one, enter them separated by comma Local minima at $z=$
Solution
Step 1 ：Find the derivative of the function: \(f'(x) = 6x^2 + 5\)
Step 2 ：Set the derivative equal to zero: \(6x^2 + 5 = 0\)
Step 3 ：Solve for x: \(x^2 = -\frac{5}{6}\)
Step 4 ：Since the square of a real number cannot be negative, there are no real solutions to this equation. Therefore, there are no critical values for this function. \(\boxed{\text{Critical value(s) = -1000}}\)
Step 5 ：To determine where the function is increasing, find where the derivative is positive. The derivative is a quadratic function that opens upwards, so it is always positive. Therefore, the function is increasing for all real numbers. \(\boxed{\text{Increasing: (-I, I)}}\)
Step 6 ：To find the local maxima of the function, find where the derivative changes from positive to negative. However, the derivative is always positive. Therefore, there are no local maxima for this function. \(\boxed{\text{Local maxima at z = -1000}}\)
Step 7 ：Similarly, to find the local minima of the function, find where the derivative changes from negative to positive. However, the derivative is always positive. Therefore, there are no local minima for this function. \(\boxed{\text{Local minima at z = -1000}}\)
