Answer :
42x⁶ - 20x⁴ + 3e² - eˣ represents f''(x) if f(x) = x⁷ - x⁵ + e³ - x + eˣ.Thus the option a) 42x⁶-20x⁴+3e²-eˣ is correct.
To find the second derivative, f''(x) ,we first find the first derivative, f'(x), using the power rule and the derivative of exponential functions.
The first derivative, f'(x), is calculated as 7x⁶ - 5x⁴ + 3e² - 1 + eˣ.
Then, we differentiate f'(x) again to find f''(x).
This yields 42x⁶ - 20x⁴ + 0 - 0 + 0, since the derivative of constants is 0 and the derivative of eˣ is eˣ.
Therefore, the second derivative of f(x) is 42x⁶ - 20x⁴ + 3e² - eˣ, which matches option (a).
This equation represents the rate of change of the rate of change of the original function f(x).
Each term in the second derivative reflects how the slope of the original function changes with respect to x.
The correct option (a) accurately represents these changes and aligns with the calculations of the second derivative.
Thus the option a) 42x⁶-20x⁴+3e²-eˣ is correct.
