Question

If y is a function of x, then y is minimum if and only if
A. $ \dfrac{dy}{dx}=0 $
B. $ \dfrac{{{d}^{2}}y}{d{{x}^{2}}}=0 $
C. $ \dfrac{{{d}^{2}}y}{d{{x}^{2}}}>0 $
D. $ \dfrac{{{d}^{2}}y}{d{{x}^{2}}}<0 $

Answer 1

Hint: Question is purely based on the concept of finding the maxima and minima using differentiation. Which says that whenever the double derivative of a function gives greater than zero then the point will give minima.

Complete step by step answer:
Moving ahead with the question,
As we are asked to find the condition of minima in the form of derivatives. As we know that according to the differentiation concept to find the minima of any function which, we had to first find the first derivative of the function and put it equal to zero. From here we will get the roots of the first derivative of the function, then we have to put these roots in the second derivative of the main function. If the values give greater than zero then at that value the function will give minimum.
So from this last statement we can say that if the double derivative of any function is greater than zero then the function will give a minimum.
So we can say for y is minimum if and only if double derivative is zero, i.e. $ \dfrac{{{d}^{2}}y}{d{{x}^{2}}}>0 $
Hence answer is $ \dfrac{{{d}^{2}}y}{d{{x}^{2}}}>0 $

So, the correct answer is “Option C”.

Note: If the double derivative of function is less than zero i.e. $ \dfrac{{{d}^{2}}y}{d{{x}^{2}}}<0 $ then the function will give minima at the point on which it gives less than zero. Moreover first derivative is represented as $ \dfrac{dy}{dx} $ and second derivative is represented as $ \dfrac{{{d}^{2}}y}{d{{x}^{2}}} $ in which function is ‘y’ in terms of ‘x’.