Find the derivative of the function f(x)=2x2+3x−5 at x=−1. Also prove that f′(0)+3f′(−1)=0.
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Step-by-Step Solution
Step 1: Find the general derivative of f(x)
To find the derivative of the function f(x)=2x2+3x−5, we apply the power rule and the constant multiple rule of differentiation. The derivative of 2x2 is 4x, the derivative of 3x is 3, and the derivative of a constant −5 is 0.
Step 2: Evaluate f′(−1)
Now we substitute x=−1 into the derivative function f′(x)=4x+3 to find the derivative at that specific point.
Step 3: Evaluate f′(0)
Similarly, we substitute x=0 into the derivative function f′(x)=4x+3 to find the derivative at x=0.
Step 4: Prove the given relation
Finally, we substitute the calculated values of f′(0) and f′(−1) into the given relation f'(0) + 3f'(-1) to verify if it equals zero.