Answer:
Global maxima are (3, 0) and (-3, 0),
Local minima is (0, -1.62)
Step-by-step explanation:
Here, the given function,
![f(x) = -0.02x^4+0.36x^2-1.62](https://tex.z-dn.net/?f=f%28x%29%20%3D%20-0.02x%5E4%2B0.36x%5E2-1.62)
Differentiating with respect to x,
![f'(x) = -0.08x^3+0.72x](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%20-0.08x%5E3%2B0.72x)
For maxima or minima,
![f'(x)=0](https://tex.z-dn.net/?f=f%27%28x%29%3D0)
![\implies-0.08x^3+0.72x=0](https://tex.z-dn.net/?f=%5Cimplies-0.08x%5E3%2B0.72x%3D0)
![-0.08x(x^2-9)=0](https://tex.z-dn.net/?f=-0.08x%28x%5E2-9%29%3D0)
![\implies x=0\text{ or }x=\pm 3](https://tex.z-dn.net/?f=%5Cimplies%20x%3D0%5Ctext%7B%20or%20%7Dx%3D%5Cpm%203)
Thus, the critical points of the function f(x) are 0, -3 and 3,
Since, f'(x) > 0 on the left side of x = -3 and f'(x) < 0 on the right side of x = -3,
⇒ x = -3 is local maxima,
Also, f(-3) = 0,
⇒ f(x) has maxima at (-3, 0),
f'(x) < 0 on the left side of x = 0 and f'(x) > 0 on the right side of x = 0,
⇒ x = 0 is the local minima,
Also, f(0) = -1.62
⇒ function f(x) has minima at (0, -1.62),
Hence, the global maxima are (3, 0) and (-3, 0),
Local minima is (0, -1.62).
f'(x) > 0 on the left side of x = 3 and f'(x) < 0 on the right side of x = 3,
⇒ x = 3 is local maxima,
Also, f(3) = 0,
⇒ function f(x) has maxima at (3, 0).
Note : function f(x) has no global minima because its end behaviour is,
![\text{As }x\rightarrow \infty ; f(x)\rightarrow -\infty](https://tex.z-dn.net/?f=%5Ctext%7BAs%20%7Dx%5Crightarrow%20%5Cinfty%20%3B%20f%28x%29%5Crightarrow%20-%5Cinfty)
![\text{As }x\rightarrow -\infty ; f(x)\rightarrow -\infty](https://tex.z-dn.net/?f=%5Ctext%7BAs%20%7Dx%5Crightarrow%20-%5Cinfty%20%3B%20f%28x%29%5Crightarrow%20-%5Cinfty)